VECTOR FIELDS

VECTOR FIELDS

Keijo Ruohonen

2013

Contents 1 1 2 5 7 9 13

I POINT. VECTOR. VECTOR FIELD

16 16 17 18 22 23 30 34 36

II MANIFOLD

38 38 41 43

III VOLUME

46 46 52 54 59

IV FORMS

1.1 Geometric Points 1.2 Geometric Vectors 1.3 Coordinate Points and Vectors 1.4 Tangent Vectors. Vector Fields. Scalar Fields 1.5 Differential Operations of Fields 1.6 Nonstationary Scalar and Vector Fields

2.1 Graphs of Functions 2.2 Manifolds 2.3 Manifolds as Loci 2.4 Mapping Manifolds. Coordinate-Freeness 2.5 Parametrized Manifolds 2.6 Tangent Spaces 2.7 Normal Spaces 2.8 Manifolds and Vector Fields

3.1 Volumes of Sets 3.2 Volumes of Parametrized Manifolds 3.3 Relaxed Parametrizations

4.1 k-Forms 4.2 Form Fields 4.3 Forms and Orientation of Manifolds 4.4 Basic Form Fields of Physical Fields

62 63 69 73 77

V GENERALIZED STOKES’ THEOREM

83 83 87 92 96 97 98

VI POTENTIAL

5.1 Regions with Boundaries and Their Orientation 5.2 Exterior Derivatives 5.3 Exterior Derivatives of Physical Form Fields 5.4 Generalized Stokes’ Theorem

6.1 Exact Form Fields and Potentials 6.2 Scalar Potential of a Vector Field in R3 6.3 Vector Potential of a Vector Field in R3 6.4 Helmholtz’s Decomposition 6.5 Four-Potential 6.6 Dipole Approximations and Dipole Potentials

i

ii 102 VII PARTIAL DIFFERENTIAL EQUATIONS 102 7.1 Standard Forms 102 7.2 Examples 107 Appendix 1: PARTIAL INTEGRATION AND GREEN’S

IDENTITIES 107 A1.1 Partial Integration 111 A1.2 Green’s Identities 111 111 113 115 117

Appendix 2: PULLBACKS AND CURVILINEAR COORDINATES

118 118 119 120 122

Appendix 3: ANGLE

A2.1 Local Coordinates A2.2 Pullbacks A2.3 Transforming Derivatives of Fields A2.4 Derivatives in Cylindrical and Spherical Coordinates

A3.1 Angle Form Fields and Angle Potentials A3.2 Planar Angles A3.3 Solid Angles A3.4 Angles in Rn

124 References 126 Index

Foreword These lecture notes form the base text for the course ”MAT-60506 Vector Fields”. They are translated from the older Finnish lecture notes for the course ”MAT-33351 Vektorikentät”, with some changes and additions. These notes deal with basic concepts of modern vector field theory, manifolds, (differential) forms, form fields, Generalized Stokes’ Theorem, and various potentials. A special goal is a unified coordinate-free physico-geometric representation of the subject matter. As a sufficient background, basic univariate calculus, matrix calculus and elements of classical vector analysis are assumed. Classical vector analysis is one of the oldest areas of mathematical analysis.1 Modelling structural strength, fluid flow, thermal conduction, electromagnetics, vibration etc. in the threespace needs generalization of the familiar concepts and results of univariate calculus. There seem to be a lot of these generalizations. Indeed, vector analysis—classical as well as modern— has been largely shaped and created by the many needs of physics and various engineering applications. For the latter, it is central to be able to formulate the problem as one where fast and accurate numerical methods can be readily applied. This generally means specifying the local behavior of the phenomenon using partial differential equations (PDEs) of a standard type, which then can be solved globally using program libraries. Here PDEs are not extensively dealt with, mainly via examples. On the other hand, basic concepts and results having to do with their derivation are emphasized, and treated much more extensively. Modern vector analysis introduces concepts which greatly unify and generalize the many scattered things of classical vector analysis. Basically there are two machineries to do this: 1

There is a touch of history in the venerable Finnish classics TALLQVIST and VÄISÄLÄ , too.

iii manifolds and form fields, and Clifford’s algebras. These notes deal with the former (the latter is introduced in the course ”Geometric Analysis”). The style, level and order of presentation of the famous textbook H UBBARD & H UBBARD have turned out to be well-chosen, and have been followed here, too, to an extent. Many tedious and technical derivations and proofs are meticulously worked out in this book, and are omitted here. As another model the more advanced book L OOMIS & S TERNBERG might be mentioned, too.

Keijo Ruohonen

”One need only know that geometric objects in spacetime are entities that exist independently of coordinate systems or reference frames.” (C.W. M ISNER & K.S. T HORNE & J.A. W HEELER : Gravitation)

Chapter 1 POINT. VECTOR. VECTOR FIELD 1.1 Geometric Points A point in space is a physico-geometric primitive, and is not given any particular definition here. Let us just say that dealing with points, lines, planes and solids is mathematically part of so-called solid geometry. In what follows points will be denoted by capital italic letters: P, Q, R, . . . and P1 , P2 , . . . , etc. The distance between the points P and Q is denoted by d(P, Q). Obviously d(P, P ) = 0, d(P, Q) = d(Q, P ) and d(P, R) ≤ d(P, Q) + d(Q, R) (triangle inequality). An open P -centered ball of radius R is the set of all points Q with d(P, Q) < R, and is denoted by B(R, P ). Further: • The point set A is open, if for each point P in it there is a number RP > 0 such that B(RP , P ) ⊆ A. In particular the empty set ∅ is open.

• The boundary ∂A of the point set A is the set of all points P such that every open ball B(R, P ) (R > 0) contains both a point of A and a point of the complement of A. In particular the boundary of the empty set is empty. A set is thus open if and only if it does not contain any of the points of its boundary. • The point set A is closed, if it contains its boundary. In particular the empty set is thus closed. Since the boundaries of a set and its complement clearly are the same, a set is closed if and only if its complement is open. • The closure1 of the point set A is the set A = A ∪ ∂A and the interior is A◦ = A − ∂A. The interior of an open set is the set itself, as is the closure of a closed set. Geometric points naturally cannot be added, subtracted, or be multiplied by a scalar (a realvalued constant). It will be remembered from basic calculus that for coordinate points these operations are defined. But in reality they are then the corresponding operations for vectors, as will be seen soon. Points and vectors are not the same thing. Note. Here and in the sequel only points, vectors and vector fields in space are explicitly dealt with. The concepts may be defined for points, vectors and vector fields in plane or real axis. For instance, an open ball in plane is an open circle, in real axis an open ball is an open finite interval, and so on. To an extent they can be defined in higher dimensions, too. 1

Do not confuse this with the complement which is often also denoted by an overbar!

1

CHAPTER 1. POINT. VECTOR. VECTOR FIELD

2

1.2 Geometric Vectors The directed line segment connecting the two points P (the initial point) and Q (the terminal −→ −→ −→ point) is denoted by P Q. Two such directed line segments P Q and RS are said to be equivalent if they can be obtained from each other by parallel transform, i.e., there is a parallel transform which takes P to R and Q to S, or vice versa. Directed line segments are thus partitioned into equivalence classes. In each equivalence class directed line segments are mutually equivalent, while those in different equivalence classes −→ −→ are not. The equivalence class containing the directed line segment P Q is denoted by hP Qi. −→ The directed line segment P Q is then a representative of the class. Each class has a representative for any given initial (resp. terminal) point. Geometric vectors can be identified with these equivalenc classes. The geometric vector −→ with a representative P Q has a direction (from P to Q) and length (the distance d(P, Q)). Since representatives of an equivalence class are equivalent via parallel transforms, direction and length do not depend on the choice of the representative. In the sequel geometric vectors will be denoted by small italic letters equipped with an overarrow: ~r, ~s, ~x . . . and ~r0 , ~r1 , . . . , etc. For convenience the zero vector ~0 will be included, too. It has no direction and zero length. The length of the vector ~r is denoted by |~r|. A vector with length = 1 is called a unit vector. A physical vector often has a specific physical unit L (sometimes also called dimension), e.g. kg/m2 /s. In this case the geometric vector ~r couples the direction of the physical action to the direction of the geometric vector—unless ~r is the zero vector—and the magnitude ~r is given in physical units L. Note that if the unit L is a unit of length, say metre, then a geometric vector ~r may be considered as a physical vector, too. A physical vector may be unitless, so that it has no attached physical unit, or has an empty unit. Physical units can be multiplied, divided and raised to powers, the empty unit has the (purely numerical) value 1 in these operations.2 Often, however, a physical vector is completely identified with a geometric vector (with a proper conversion of units). In the sequel we just generally speak about vectors. Vectors have the operations familiar from basic courses of mathematics. We give the geometric definitions of these in what follows. Geometrically it should be quite obvious that they are well-defined, i.e., independent of choices of representatives. −→ • The opposite vector of the vector ~r = hP Qi is the vector −→ −~r = hQP i.

In particular −~0 = ~0. The unit of a physical vector remains the same in this operation. • The sum of the vectors

−→ ~r = hP Qi

−→ and ~s = hQRi

(note the choice of the representatives) is the vector −→ ~r + ~s = hP Ri. In particular we define ~r + ~0 = ~0 + ~r = ~r See e.g. G IBBINGS , J.C.: Dimensional Analysis. Springer–Verlag (2011) for many more details of dimensional analysis. 2


3

and ~r + (−~r ) = (−~r ) + ~r = ~0. Only vectors sharing a unit can be physically added, and the unit of the sum is this unit. Addition of vectors is commutative and associative, i.e. ~r + ~s = ~s + ~r

and ~r + (~s + ~t ) = (~r + ~s ) + ~t.

These are geometrically fairly obvious. Associativity implies that long sums may be parenthesized in any (correct) way, or written totally without parenteheses, without the result changing. The difference of the vectors ~r and ~s is the vector ~r − ~s = ~r + (−~s ). For physical vectors the units again should be the same in this operation. −→ • If ~r = hP Qi is a vector and λ a positive scalar, then λ~r is the vector obtained as follows: – Take the ray starting from P through the point Q. – In this ray find the point R whose distance from P is λ|~r |. −→ – Then λ~r = hP Ri. In addition it is agreed that λ~0 = ~0 and 0~r = ~0. This operation is multiplication of vector by scalar. Defining further (−λ)~r = −(λ~r ) we get multiplication by a negative scalar. Evidently 1~r = ~r

,

(−1)~r = −~r

,

2~r = ~r + ~r ,

etc.

If the physical scalar λ and physical vector ~r have their physical units, the unit of λ~r is their product. With a bit of work the following laws of calculation can be (geometrically) verified: λ1 (λ2~r) = (λ1 λ2 )~r , (λ1 + λ2 )~r = λ1~r + λ2~r and λ(~r1 + ~r2 ) = λ~r1 + λ~r2 , where λ, λ1 , λ2 are scalars and ~r, ~r1 , ~r2 are vectors. Division of a vector ~r by a scalar λ 6= 0 is multiplication of the vector by the inverse 1/λ, denoted by ~r/λ. A frequently occurring multiplication by scalar is normalization of a vector where a vector ~r 6= ~0 is divided by its length. The result ~r/|~r | is a unit vector (and unitless).

−→ −→ • The angle ∠(~r, ~s ) spanned by the vectors ~r = hP Qi and ~s = hP Ri (note the choice −→ −→ of representatives) is the angle between the directed line segments P Q and P R given in radians in the interval [0, π] rad. It is obviously assumed here that ~r, ~s 6= ~0. Moreover an angle is always unitless, the radian is not a physical unit.

CHAPTER 1. POINT. VECTOR. VECTOR FIELD • The distance of the vectors

4

−→ −→ ~r = hP Qi and ~s = hP Ri

(note the choice of representatives) is d(~r, ~s ) = d(Q, R) = |~r − ~s |. In particular d(~r, ~0 ) = |~r |. This distance also satisfies the triangle inequality d(~r, ~s ) ≤ d(~r, ~t ) + d(~t, ~s ). • The dot product (or scalar product) of the vectors ~r 6= ~0 and ~s 6= ~0 is ~r • ~s = |~r ||~s | cos ∠(~r, ~s ). In particular ~r • ~0 = ~0 • ~r = 0 and ~r • ~r = |~r |2 . Dot product is commutative, i.e. and bilinear, i.e.

~r • ~s = ~s • ~r, ~r • (λ~s + η~t ) = λ(~r • ~s ) + η(~r • ~t ),

where λ and η are scalars. Geometrically commutativity is obvious. Bilinearity on the other hand requires a somewhat complicated geometric proof. Using coordinates makes bilinearity straightforward. The unit of the dot product of physical vectors is the product of their units. Geometrically, if ~s is a (unitless) unit vector, then ~r • ~s = |~r | cos ∠(~r, ~s ) is the (scalar) projection of ~r on ~s. (The projection of a zero vector is of course always zero.) • The cross product (or vector product) of the vectors ~r and ~s is the vector ~r × ~s given by the following. First, if ~r or ~s is ~0, or ∠(~r, ~s ) is 0 or π, then ~r × ~s = ~0. Otherwise ~r × ~s is the unique vector ~t satisfying – |~t | = |~r ||~s | sin ∠(~r, ~s ), π – ∠(~r, ~t ) = ∠(~s, ~t ) = , and 2 – ~r, ~s, ~t is a right-handed system of vectors. Cross product is anticommutative, i.e. ~r × ~s = −(~s × ~r ), and bilinear, i.e. ~r × (λ~s + η~t ) = λ(~r × ~s ) + η(~r × ~t ),

where λ and η are scalars. Geometrically anticommutatitivy is obvious, handidness changes. Bilinearity again takes a complicated geometric proof, but is fairly easily seen using coordinates.


5

Cross product is an information-dense operation, involving lengths of vectors, angles and handidness. It is easily handled using coordinates, too. Geometrically |~r × ~s | = |~r ||~s | sin ∠(~r, ~s ) is the area of the parallelogram with lengths of sides |~r | and |~s | and spanning angle ∠(~r, ~s ). If ~r and ~s are physical vectors, then the unit of the cross product ~r × ~s is the product of their units. • Combining these products we get the scalar triple product ~r • (~s × ~t ) and the vector triple products (~r × ~s ) × ~t and ~r × (~s × ~t ). There being no danger of confusion, the scalar product is usually written without parentheses as ~r • ~s × ~t. Scalar triple product is cyclically symmetric, i.e.

~r • ~s × ~t = ~s • ~t × ~r = ~t • ~r × ~s. By this and commutativity of scalar product, the operations • and × can be interchanged, i.e. ~r • ~s × ~t = ~r × ~s • ~t. Geometrically it is easily noted that the scalar triple product ~r • ~s × ~t is the volume of the parallelepiped with edges incident on the vertex P being the vectors −→ ~r = hP Ri ,

−→ −→ ~s = hP Si and ~t = hP T i,

with a positive sign if ~r, ~s, ~t is a right-handed system, and a negative sign otherwise. (As special cases situations where the scalar triple product is = 0 should be included, too.) Cyclic symmetry follows geometrically immediately from this observation. The triple vector product expansions (also known as Lagrange’s formulas) are (~r × ~s ) × ~t = (~r • ~t )~s − (~s • ~t )~r and ~r × (~s × ~t ) = (~r • ~t )~s − (~r • ~s )~t. These are somewhat difficult to prove geometrically, proofs using coordinates are easier. ~ Exactly as for points we can define an ~r-centered open ball B(R, ~r ) of radius R for vectors, open and closed sets of vectors, and boundaries, closures and interiors of sets of vectors, but a geometric intuition is then not as easily obtained.

1.3 Coordinate Points and Vectors In basic courses on mathematics points are thought of as coordinate points, i.e. triples (a, b, c) of real numbers. In the background there is then an orthonormal right-handed coordinate system with its axes and origin. Coordinate points will be denoted by small boldface letters: r, s,


6

x, . . . and r0 , r1 , . . . , etc. The coordinate point corresponding to the origin of the system is 0 = (0, 0, 0). A coordinate system is determined by the corresponding coordinate function which maps geometric points to the triples of R3 . We denote coordinate functions by small boldface Greek letters, and their components by the corresponding indexed letters. If the coordinate function is κ, then the coordinates of the point P are κ(P ) = κ1 (P ), κ2 (P ), κ3(P ) .

A coordinate function is bijective giving a one-to-one correspondence between the geometric space and the Euclidean space R3 . Distances are given by the familiar Euclidean norm of R3 . If κ(P ) = (x1 , y1 , z1 )

and κ(Q) = (x2 , y2 , z2 ),

then

p d(P, Q) = κ(P ) − κ(Q) = (x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2 .

A coordinate function κ also gives a coordinate representation for vectors. The coordinate −→ version of the vector ~r = hP Qi is   κ (Q) − κ (P ) 1 1 T κ(~r ) = κ(Q) − κ(P ) =  κ2 (Q) − κ2 (P )  . κ3 (Q) − κ3 (P )

Note that this is a column array. As is easily seen, this presentation does not depend on choices of the representative directed line segments. In particular the zero vector has the representation −→ −→ κ(~0 ) = (0, 0, 0)T = 0T . Also the distance of the vectors ~r = hP Qi and ~s = hP Ri (note the choice of representatives) is obtained from the Euclidean norm of R3 :

d(~r, ~s ) = κ(~r ) − κ(~s ) = κ(Q) − κ(R) = d(Q, R).

And then |~r | = d(~r, ~0) = κ(~r ) . In the sequel also coordinate representations, or coordinate vectors, will be denoted by small boldface letters, but it should be remembered that a coordinate vector is a column vector. Certain coordinate vectors have their traditional notations and roles:         1 0 0 x i =  0  , j =  1  , k =  0  and r =  y  . 0 0 1 z The vectors i, j, k are the basis vectors and the vector r is used as a generic variable vector. In the background there of course is a fixed coordinate system and a coordinate function. The row array versions of these are also used as coordinate points. Familiar operations of coordinate vectors now correspond exactly to the geometric vector operations in the previous section. Let us just recall that if     a1 a2 κ(~r ) = r =  b1  and κ(~s ) = s =  b2  , c1 c2 then

~r • ~s = r • s = a1 a2 + b1 b2 + c1 c2

CHAPTER 1. POINT. VECTOR. VECTOR FIELD and

7



 b1 c2 − b2 c1 κ(~r × ~s ) = r × s =  c1 a2 − c2 a1  . a1 b2 − a2 b1

The latter is often given as the more easily remembered formal determinant i a1 a2 j b1 b2 , k c1 c2

to be expanded along the first column. A coordinate transform changes the coordinate function. If κ and κ∗ are two available coordinate functions, then they are connected via a coordinate transform, that is, there is a 3 × 3 orthogonal matrix3 Q and a coordinate vector b such that κ∗ (P ) = κ(P )Q + b and κ(P ) = κ∗ (P )QT − bQT . −→ The coordinate representation of a vector ~r = hP Qi is transformed similarly: κ∗ (~r ) = κ∗ (Q) − κ∗ (P )

T

= QT κ(Q) − κ(P )

and

T = κ(Q)Q + b − κ(P )Q − b

T

= QT κ(~r )

κ(~r ) = Qκ∗ (~r ). Note that b is the representation of the origin of the ”old” coordinate system in the ”new” system, and that the columns of QT are the representations of the basis vectors i, j, k of the ”old” coordinate system in the ”new” system. Similarly −bQT is the representation of the origin of the ”new” coordinate system in the ”old” system, and the columns of Q are the representations of the ”new” basis vectors i∗ , j∗ , k∗ in the ”old” system.

1.4 Tangent Vectors. Vector Fields. Scalar Fields −→ Geometrically a tangent vector4 is simply a directed line segment P Q, the point P is its point of action. It is however easier to think a tangent vector (especially a physical one) as a pair [P, ~r ] where P is the point of action and ~r is a vector. It is then easy to apply vector operations to tangent vectors: just operate on the vector component ~r. If the result is a vector, it may be thought of as a tangent vector, with the original (joint) point of action, or as just a vector without any point of action. Moreover, in the pair formulation, a coordinate representation is simply obtained using a coordinate function κ: κ [P, ~r ] = κ(P ), κ(~r ) . Often in the pair [P, ~r ] we consider ~r as a physical vector operating in the point P . 3

Here and in the sequel matrices are denoted by boldface capital letters. Since handidness needs to be preserved, we must have here det(Q) = 1. Recall that orthogonality means that Q−1 = QT which implies that det(Q)2 = 1. 4 Thus called because it often literally is a tangent.


8

If the point of action is clear from the context, or is irrelevant, it is often omitted and only the vector component of the pair is used, usually in a coordinate representation. This is what we will mostly do in the sequel. A vector field is a function mapping a point P to a tangent vector P, F~ (P ) (note the point of action). Mostly we denote this just by F~ . A vector field may not be defined for all points of the geometric space, i.e., its domain of definition may be smaller. In the coordinate representation given by the coordinate function κ we denote r = κ(P ) and F(r) = κ F~ (P ) ,

thus coordinate vector fields are denoted by capital boldface letters. Note that in the coordinate transform r∗ = rQ + b (i.e. κ∗ (P ) = κ(P )Q + b) the vector field F = κ(F~ ) (that is, its representation) is transformed to the the field F∗ = κ∗ (F~ ) given by the formula F∗ (r∗ ) = QT F (r∗ − b)QT .

A vector field may of course be defined in fixed coordinates in one way or another, and then taken to other coordinate systems using the transform formula. On the other hand, definition of a physico-geometric vector field cannot possibly depend on any coordinate system, the field exists without any coordinates, and will automatically satisfy the transform formula. A coordinate vector field is the vector-valued function of three arguments familiar from basic courses of mathematics   F1 (r) F(r) =  F2 (r)  , F3 (r)

with its components. Thus all operations and concepts defined for these apply, limits, continuity, differentiability, integrals, etc. A scalar field is a function f mapping a point P to a scalar (real number) f (P ), thus scalar fields are denoted by italic letters, usually small. In the coordinate representation we denote r = κ(P ) and just f (r) (instead of the correct f κ−1 (r) ). In the coordinate transform r∗ = rQ + b

(i.e. κ∗ (P ) = κ(P )Q + b)

a scalar field f is transformed to the scalar field f ∗ given by the formula f ∗ (r∗ ) = f (r∗ − b)QT .

A scalar field, too, can be defined in fixed coordinates, and then transformed to other coordinate systems using the transform formula. But a physico-geometric scalar field exists without any coordinate systems, and will automatically satisfy the transform formula. Again a coordinate scalar field is the real-valued function of three arguments familiar from basic courses of mathematics. Thus all operations and concepts defined for these apply, limits, continuity, differentiability, integrals, etc. An important observation is that all scalar and vector products in the previous section applied to vector and scalar fields will again be fields, e.g. a scalar field times a vector field is a vector field.


9

1.5 Differential Operations of Fields Naturally, partial derivatives cannot be defined for physico-geometric fields, since they are intrinsically connected with a coordinate system. In a coordinate representation partial derivatives can be defined, as was done in the basic courses. In this way we get partial derivatives of a scalar field f , its derivative f′ = and gradient

∂f ∂f ∂f , , ∂x ∂y ∂z  ∂f  ∂x     ∂f   , =  ∂y    ∂f  

grad(f ) = ∇f = f ′T

∂z



 F1 and the derivative or Jacobian (matrix) of a vector field F =  F2  F3 ∂F1    ∂x  F1′  ∂F2  ′ ′  F =  F2  =   ∂x  F3′  ∂F3 ∂x 

∂F1 ∂y ∂F2 ∂y ∂F3 ∂y

 ∂F1 ∂z   ∂F2  . ∂z   ∂F3  ∂z

Using the chain rule5 we get the transforms of the derivatives in a coordinate transform r = rQ + b: ′ f ∗ ′ (r∗ ) = f (r∗ − b)QT = f ′ (r∗ − b)QT Q ∗

and

F∗′ (r∗ ) = QT F (r∗ − b)QT

′

= QT F′ (r∗ − b)QT Q.

Despite partial derivatives depending on the coordinate system used, differentiability itself is coordinate-free: If a field has partial derivates in one coordinate system, it has them in any other coordinate system, too. This is true for second order derivatives as well. And it is true for continuity: Continuity in one coordinate system implies continuity in any other system. And finally it is true for continuous differentiability: If a field has continuous partial derivatives (first or second order) in one coordinate system, it has them in any other system, too. All this follows from the transform formulas.

5

Assuming f and g differentiable, the familiar univariate chain rule gives the derivative of the composite function: ′ f g(x) = f ′ g(x) g ′ (x).

More generally, assuming F and G continuously differentiable (and given as column arrays), we get the derivative of the composite function as ′ = F′ G(r)T G′ (r). F G(r)T

The arguments are here thought of as row arrays. The rule is valid in higher dimensions, too.


10

The common differential operations for fields are the gradient (nabla) of a scalar field f , and its Laplacian ∂2f ∂2f ∂f 2 ∆f = ∇ • (∇f ) = ∇2 f = + + , ∂x2 ∂y 2 ∂z 2   F1  and for a vector field F = F2  its divergence F3 div(F) = ∇ • F = and curl

∂F1 ∂F2 ∂F3 + + ∂x ∂y ∂z

 ∂F3 ∂F2 ∂ i − F 1  ∂y  ∂z ∂x    ∂F1 ∂F3  ∂   − = curl(F) = ∇ × F =  j F 2 .  ∂z ∂x ∂y    ∂F2 ∂F1  ∂ − k F3 ∂x ∂y ∂z (As the cross product, the curl can be given as a formal determinant.) As will be verified shortly, gradient, divergence and curl are coordinate-free. Thus ∇ • F can be interpreted as a scalar field, and, as already indicated by the column array notation, ∇f and ∇ × F as vector fields. For the gradient coordinate-freeness is physically immediate. It will be remembered that the direction of the gradient is the direction of fastest growth for a scalar field, and its length is this speed of growth (given as a directional derivative). For divergence and curl the situation is not at all as obvious. It follows from the coordinate-freenes of the gradient that the directional derivative of a scalar field f in the direction n (a unit vector) 

∂f = n • ∇f ∂n is also coordinate-free and thus a scalar field. The Laplacian may be applied to a vector field as well as follows:   ∆F1 ∆F =  ∆F2  . ∆F3

This ∆F is coordinate-free, too, and can be interpreted as a vector field. A central property, not to be forgotten, is that all these operations are linear, in other words, if λ1 and λ2 are constant scalars, then e.g. ∇(λ1 f + λ2 g) = λ1 ∇f + λ2 ∇g

and

∇ • (λ1 F + λ2 G) = λ1 ∇ • F + λ2 ∇ • G , etc. The following notational expression appears often: G • ∇ = G1

∂ ∂ ∂ + G2 + G3 , ∂x ∂y ∂z


11

where G = (G1 , G2 , G3 )T is a vector field. This is interpreted as an operator applied to a scalar field f or a vector field F = (F1 , F2 , F3 )T as follows: (G • ∇)f = G • (∇f ) = G1

∂f ∂f ∂f + G2 + G3 ∂x ∂y ∂z

and (taking F1 , F2 , F3 to be scalar fields) 

 (G • ∇)F1 (G • ∇)F =  (G • ∇)F2  = F′ G. (G • ∇)F3

These are both coordinate-free and hence fields. Coordinate-freeness of (G • ∇)F follows from the nabla rules below (or from the coordinate-freeness of F′ G). Let us tabulate the familiar nabla-calculus rules: (i) ∇(f g) = g∇f + f ∇g (ii) ∇

1 1 = − 2 ∇f f f

(iii) ∇ • (f G) = ∇f • G + f ∇ • G (iv) ∇ × (f G) = ∇f × G + f ∇ × G (v) ∇ • (F × G) = ∇ × F • G − F • ∇ × G (vi) ∇ × (F × G) = (G • ∇)F − (∇ • F)G + (∇ • G)F − (F • ∇)G (vii) ∇(F • G) = (G • ∇)F − (∇ × F) × G − (∇ × G) × F + (F • ∇)G In matrix notation ∇(F • G) = F′ T G + G′ T F. (viii) (∇ × F) × G = (F′ − F′ T )G (ix) ∇ • (∇ × F) = 0 (x) ∇ × ∇f = 0 (xi) ∇ × (∇ × F) = ∇(∇ • F) − ∆F

(so-called double-curl expansion)

(xii) ∆(f g) = f ∆g + g∆f + 2∇f • ∇g In formulas (ix), (x), (xi) we assume F and f are twice continuously differentiable. These formulas are all symbolical identities, and can be verified by direct calculation, or e.g. using the Maple symbolic computation program.


12

Let us, as promised, verify coordinate-freeness of the operators. In a coordinate transform r∗ = rQ + b we denote the nabla in the new coordinates by ∇∗ . Coordinate-freeness for the basic operators then means the following: 1. ∇∗ f (r∗ − b)QT = QT ∇f (r) (gradient)

Subtracting b and multiplying by QT we move from the new coordinates r∗ to the old ones, get the value of f , and then the gradient in the new coordinates. The result must be the same when the gradient is obtained in the old coordinates and then transformed to the new ones by multiplying by QT . 2. ∇∗ • QT F (r∗ − b)QT = ∇ • F(r) (divergence) Subtracting b and multiplying by QT we move from the new coordinates r∗ to the old ones, get F, transform the result to the new coordinates by multiplying by QT , and get the divergence using the new coordinates. The result must remain the same when the divergence is obtained in the old coordinates. 3. ∇∗ × QT F (r∗ − b)QT = QT ∇ × F(r) (curl)

Subtracting b and multiplying by QT we move from the new coordinates r∗ to the old ones, get F, transform the result to the new coordinates by multiplying by QT , and get the curl using the new coordinates. The result must be the same when the curl is obtained in the old coordinates and then transformed to the new ones by multiplying by QT .

Theorem 1.1. Gradient, divergence, curl and Laplacian are coordinate-free. Furthermore, if F and G are vector fields (and thus coordinate-free), then so is (G • ∇)F. Proof. By the above T f ∗ ′ (r∗ ) = ∇f (r) Q

and F∗ ′ (r∗ ) = QT F′ (r)Q.

This immediately gives formula 1. since

∇∗ f ∗ (r∗ ) = f ∗ ′ (r∗ )T = QT ∇f (r). To show formula 2. we use the trace of the Jacobian. Let us recall that the trace of a square matrix A, denoted trace(A), is the sum of the diagonal elements. A nice property of trace6 is that if the product matrix AB is square—whence BA is square, too—then trace(AB) = trace(BA). Since trace(F′ ) =

6

∂F1 ∂F2 ∂F3 + + = ∇ • F, ∂x ∂y ∂z

Denoting A = (aij ) (an n × m matrix), B = (bij ) (an m × n matrix), AB = (cij ) and BA = (dij ) we have trace(AB) =

n X

k=1

ckk =

n X m X k=1 l=1

akl blk =

m X n X l=1 k=1

blk akl =

m X l=1

dll = trace(BA).


13

formula 2. follows: ∇∗ • F∗ (r∗ ) = trace F∗ ′ (r∗ ) = trace QT F′ (r)Q = trace QQT F′ (r) (take B = Q) = trace F′ (r) = ∇ • F(r).

To prove formula 3. we denote the columns of Q by q1 , q2 , q3 . Let us consider the first component of the curl ∇∗ ×F∗ (r∗ ). Using the transform formula for the Jacobian, nabla formula (viii) and rules for the scalar triple product we get ∗

∗

∗

∇ × F (r )

1

∂F3∗ ∂F2∗ = − ∂y ∗ ∂z ∗

= qT3 F′ (r)q2 − qT2 F′ (r)q3

(because F∗ ′ = QT F′ Q)

= qT3 F′ (r)q2 − qT3 F′ (r)T q2

′ (qT 2 F q3 is a scalar)

= qT3 F′ (r) − F′ (r)T q2 = q3 • ∇ × F(r) × q2 = q2 • q3 × ∇ × F(r) = q2 × q3 • ∇ × F(r) = q1 • ∇ × F(r)

(extract the factors qT 3 and q2 )

= QT ∇ × F(r)

1

(formula (viii)) (cyclic symmetry) (interchange • and ×) (here q2 × q3 = q1 )

.

Note that q2 × q3 = q1 since the new coordinate system must be right-handed, too. The other components are dealt with similarly. Coordinate-freeness of the Laplacian follows directly from that of gradient and divergence for scalar fields, and for vector fields from formula (xi). Adding formulas (vi) and (vii) on both sides we get an expression for (G • ∇)F: 1 ∇ × (F × G) + (∇ • F)G − (∇ • G)F + ∇(F • G) 2 + (∇ × F) × G + (∇ × G) × F .

(G • ∇)F =

All six terms on the right hand side are coordinate-free and thus vector fields. The left hand side (G • ∇)F then also is coordinate-free and a vector field. (This is also easily deduced from the matrix form (G • ∇)F = F′ G, but the formula above is of other interest, too!)

1.6 Nonstationary Scalar and Vector Fields Physical fields naturally are often time-dependent or dynamical, that is, in the definition of the field a time variable t must appear. A scalar field is then of the form f (P, t) and a vector field of the form F~ (P, t). (The point of action is omitted here, even though that, too, may be time-dependent.) In a coordinate representation these forms are respectively f (r, t) and F(r, t). Time-dependent fields are called nonstationary, and time-independent fields are called stationary.


14

From the coordinate representation, interpreted as a function of the four variables x, y, z, t, we again get the concepts continuity, differentiability, etc., familiar from basic courses, also for the time variable t. In a coordinate transform r∗ = rQ + b the time variable is not transformed, i.e. f ∗ (r∗ , t) = f (r∗ − b)QT , t and F∗ (r∗ , t) = QT F (r∗ − b)QT , t .

Thus for the time derivatives we get the corresponding transform formulas ∂i ∗ ∗ ∂i ∗ T f (r , t) = f (r − b)Q , t and ∂ti ∂ti

i ∂i ∗ ∗ T ∂ ∗ T F (r , t) = Q F (r − b)Q , t , ∂ti ∂ti

which shows that they are fields. In addition to the familiar partial derivative rules we get for the time derivatives e.g. the following rules, which can be verified by direct calculation:

(1)

∂F ∂G ∂ (F • G) = •G+F• ∂t ∂t ∂t

(2)

∂ ∂F ∂G (F × G) = ×G+F× ∂t ∂t ∂t

(3)

∂ ∂f ∂F (f F) = F+f ∂t ∂t ∂t

(4)

∂ ∂F ∂G ∂H (F • G × H) = •G×H+F• ×H+F•G× ∂t ∂t ∂t ∂t

∂G ∂F ∂ ∂H (5) F × (G × H) = × (G × H) + F × ×H +F× G× ∂t ∂t ∂t ∂t Another kind of time dependence in a coordinate representation is obtained by allowing a moving coordinate system (e.g. a rotating one, as in a carousel). If the original representation in a fixed coordinate system is f (r) (a scalar field) or F(r) (a vector field), then at time t we have a coordinate transform r∗ (t) = rQ(t) + b(t) and the representations of the fields are f ∗ (r∗ , t) = f

r∗ (t) − b(t) Q(t)T and

F∗ (r∗ , t) = Q(t)T F r∗ (t) − b(t) Q(t)T .

Note that now the fields are stationary, time-dependence is only in the coordinate representation and is a consequence of the moving coordinate system.


15

Similarly for nonstationary fields f (r, t) and F(r, t) in a moving coordinate system we get the representations f ∗ (r∗ , t) = f r∗ (t) − b(t) Q(t)T , t and F∗ (r∗ , t) = Q(t)T F r∗ (t) − b(t) Q(t)T , t .

Now part of the time-dependence comes from the time-dependence of the fields, part from the moving coordinate system.

”The intuitive picture of a smooth surface becomes analytic with the concept of a manifold. On the small scale a manifold looks like Euclidean space, so that infinitesimal operations like differentiation may be defined on it.’ (WALTER T HIRRING : A Course in Mathematical Physics)

Chapter 2 MANIFOLD 2.1 Graphs of Functions The graph of a function f : A → Rm , where A ⊆ Rk is an open set, is r, f(r) r ∈ A ,

a subset of Rk+m . The graph is often denoted—using a slight abuse of notation—as follows: s = f(r)

(r ∈ A).

Here r contains the so-called active variables and s the so-called passive variables. Above active variables precede the passive ones in the order of components. In a graph we also allow a situation where the variables are scattered. A graph is smooth,1 if f is continuously differentiable in its domain of definition A. In the sequel we only deal with smooth graphs. Note that a graph is specifically defined using a coordinate system and definitely is not coordinate-free. A familiar graph is the graph of a real-valued z univariate function f in an open interval (a, b), i.e., 2 the subset of R consisting of the pairs z = f(x,y) x, f (x) (a < x < b), the curve y = f (x). Another is the graph of a realvalued bivariate function f , i.e., the surface z = f (x, y) ((x, y) ∈ A)

y A

3

in the space R (see the figure on the right). Not all x curves or surfaces are graphs, however, e.g. circles (x,y) and spheres are not (and why not?). The most common dimensions k and m are of course the ones given by physical position and time, that is 1, 2 and 3, whence k + m is 2, 3 or 4. On the other hand, degrees of freedom in mechanical systems etc. may lead to some very high dimensions As a limiting case we also allow m = 0. In Rm = R0 there is then only one element (the so-called empty vector ()). Moreover, then Rk+m = Rk , thus all variables are active and the graph of the function f : A → Rm is A. Open subsets of the space are thus always graphs, and agreed to be smooth, too. 1

In some textbooks smoothness requires existence of continuous derivatives of all orders.

16

CHAPTER 2. MANIFOLD

17

Similarly we allow k = 0. Then f has no arguments, and so it is constant, and the graph consists of one point.2 Again it is agreed that such graphs are also smooth. In what follows we will need inverse images of sets. For a function g : A → B the inverse image of the set C is the set g−1 (C) = r g(r) ∈ C .

Note that this has little to do with inverse functions, indeed the function g need not have an inverse at all, and C need not be included in B. For a continuous function defined in an open set A the inverse image of an open set is always open.3 This implies an important property of graphs: Theorem 2.1. If a smooth graph of a function is intersected by an open set, then the result is either empty or a smooth graph.

Proof. This is clear if the intersection is empty, and also if k = 0 (the intersection is a point) or m = 0 (intersection of two open sets is open). Otherwise the intersection of the graph s = f(r)

(r ∈ A)

and the open set C in Rk+m is the graph s = f(r)

(r ∈ D),

where D is the inverse image of C for the continuous4 function g(r) = r, f(r) .

2.2 Manifolds A subset M of Rn is a k-dimensional manifold, if it is locally a smooth graph of some function of k variables.5 ”Locally” means that for every point p of M there is an open set Bp of Rn containing the point p such that M ∩ Bp is the graph of some function fp of some k variables. For different points p the set Bp may be quite different, the active variables chosen in a different way, always numbering k, however, and the function fp may be very different. The functions fp are called charts, and the set of all charts is called an atlas. Often a small atlas is preferable. Example. A circle of radius R centered in the origin is a 1-dimensional manifold of R2 , since (see the figure below) its each point is in an open arc delineated by either black dots or white dots, and these are smooth graphs of the functions p √ y = ± R2 − x2 and x = ± R2 − y 2 (atlas) in certain open intervals. 2

Here we may take A to be the whole space R0 , an open set. This in fact is a handy definition of continuity. Continuity of g in the point r0 means that taking C to be an arbitrarily small g(r0 )-centered open ball, there is in A a (small) r0 -centered open ball which g maps to C. 4 It is not exactly difficult to see that if f is continuous in the point r0 , then so is g, because

g(r) − g(r0 ) 2 = kr − r0 k2 + f (r) − f (r0 ) 2 . 3

5

There are several definitions of manifolds of different types in the literature. Ours is also used e.g. in H UB & H UBBARD and N IKOLSKY & VOLOSOV. They are also often more specifically called ”smooth manifolds” or ”differentiable manifolds”. With the same underlying idea so-called abstract manifolds can be defined. BARD

CHAPTER 2. MANIFOLD

18

In a similar fashion the sphere x2 + y 2 + z 2 = R2 is seen to be a 2-dimensional manifold of R3 . Locally it is a smooth graph of one of the six functions p √ x = ± R2 − y 2 − z 2 , y = ± R2 − x2 − z 2

y

x

and

p z = ± R2 − x2 − y 2

(atlas) in properly chosen open sets.

Of course, each smooth graph itself is a manifold, in particular each open subset of Rn is its n-dimensional manifold, and each single point is a 0-dimensional manifold. If a space curve is a smooth graph, say of the form (y, z) = f1 (x), f2 (x) (a < x < b),

where f1 and f2 are continuously differentiable, then it will be a 1-dimensional manifold of R3 . Also the surface z = f (x, y) ((x, y) ∈ A)

is a manifold of R3 if f is continuously differentiable. On the other hand, e.g. the graph of the absolute value function y = |x| is not smooth and therefore not a manifold of R2 . A manifold can always be restricted to be more local. As an immediate consequence of Theorem 2.1 we get Theorem 2.2. If a k-dimensional manifold of Rn is intersected by an open set, then the result is either empty or a k-dimensional manifold. Note. Why do we need manifolds? The reason is that there is an unbelievable variety of loosely defined curves and surfaces, and there does not seem to be any easy general global method to deal with them. There are continuous curves which fill a square or a cube, or which intersect themselves in all their points, continuous surfaces having normals in none of their points, etc. The only fairly easy way to grasp this phenomenon is to localize and restrict the concepts sufficiently far, at the same time preserving applicability as much as possible. Finding and proving global results is then a highly demanding and challenging area of algebro-topological mathematics in which many Fields Medals have been awarded.

2.3 Manifolds as Loci One way to define a manifold is to use loci. A locus is simply a set of points satisfying some given conditions. For instance, the P -centered circle of radius R is the locus of points having distance R from P . As was noted, it is a manifold. In general a locus is determined via a coordinate representation, and the conditions are given as mathematical equations. A condition then is of the form F(r, s) = 0, where r has dimension k, s has dimension n − k, and F is a n − k-dimensional function of n variables. The locus of points satisfying the condition is then the set of points (r, s) in Rn determined as solutions of the equation, solving for s. As indicated by the notation used, r is purported to be active and s passive. Even though here active variables appear before the passive ones in the component order, active variables may be scattered, roles of variables may differ locally, actives changing to passives, etc.

CHAPTER 2. MANIFOLD

19

Example. A circle and a sphere are loci of this kind when we set the conditions F (x, y) = R2 − x2 − y 2 = 0 and F (x, y, z) = R2 − x2 − y 2 − z 2 = 0

(centered in the origin and having radius R). In the circle one of the variables is always active, in the sphere two of the three variables. Not all loci are manifolds. For instance, the locus of points of R2 determined by the condition y − |x| = 0 is not, and neither is the locus of points satisfying the condition y 2 − x2 = 0. The former is not smooth in the origin, and the latter is not a graph of any single function in the origin (but rather of two functions: y = ±x). Actually the condition (y − x)2 = 0 does not determine a proper manifold either. The locus is the line y = x, but counted twice! So, in the equation F(r, s) = 0 surely F then should be continuously differentiable, and somehow uniqueness of solution should be ensured, too, at least locally. In classical analysis there is a result really taylor-made for this, the so-called Implicit Function Theorem, long known and useful in many contexts. Here it is used to make the transition from local loci6 to graphs. It should be mentioned that in the literature there are many versions of the theorem,7 we choose just one of them. Implicit Function Theorem. Assume that the function F : S → Rn−k , where 0 ≤ k < n, satisfies the following conditions: 1. The domain of definition S is an open subset of Rn . 2. F is continuously differentiable. 3. F′ is of full rank in the point p0 , i.e., the rows of F′ (p0 ) are linearly independent (whence also some n − k columns are also linearly independent). 4. F(p0 ) = 0 and the n − k columns of F′ (p0 ) corresponding to the variables in s are linearly independent. Denote by r variables other than the ones in s. By changing the order of variables, if necessary, we may assume that p = (r, s), and especially p0 = (r0 , s0 ). Denote further the derivative of F with respectto the variables in r by F′r , and with respect to the variables in s by F′s . (Thus F′ = F′r F′s in block form.) Then there is an open subset B of Rk containing the point r0 , and a uniquely determined function f : B → Rn−k such that 6

This is not pleonasm, although it might seem to be since ’local’ could be constructed to mean ’relating to a locus’ etc. 7 See e.g. K RANTZ , S.G. & PARKS , H.R.: The Implicit Function Theorem. History, Theory, and Applications. Birkhäuser (2012).

CHAPTER 2. MANIFOLD

20

(i) the graph of f is included in S, (ii) p0 = r0 , f(r0 ) , (iii) F r, f(r) = 0 in B,

(iv) f is continuously differentiable, the matrix F′s r, f(r) is nonsingular in B, and f ′ (r) = −F′s r, f(r)

(this only for k > 0).

−1

F′r r, f(r)

Proof. The proof is long and tedious, and is omitted here, see e.g. A POSTOL or H UBBARD & H UBBARD or N IKOLSKY & VOLOSOV. The case k = 0 is obvious, however. Then r0 is the empty vector and f is the constant function p0 . The derivative of f in item (iv) is obtained by implicit derivation, i.e., applying the chain rule to the left hand side of the identity F r, f(r) = 0 in B, and then solving for f ′ (r) the obtained equation F′r r, f(r) + F′s r, f(r) f ′ (r) = O.

Using the Implicit Function Theorem (and Theorem 2.1) we immediately get definition of manifolds using local loci: Corollary 2.3. If for any point p0 in the subset M of Rn there is a subset S of of Rn and a function F : S → Rn−k such that the conditions 1.– 4. in the Implicit Function Theorem are satisfied, and the locus condition F(p) = 0 defines the set M ∩ S, then M is a k-dimensional manifold of Rn . The converse holds true, too, i.e., all manifolds are local loci: Theorem 2.4. If M is a k-dimensional manifold of Rn and k < n, then for each point p of M there is a set S and a function F : S → Rn−k such that the conditions 1.— 4. of the Implicit Function Theorem are satisfied and the locus condition F(p) = 0 defines the set M ∩ S. Proof. Let us just see the case k > 0. (The case where k = 0 is similar—really a special case.) If M is a k-dimensional manifold of Rn , then locally in some open set containing the point p0 it is the graph of some continuously differentiable function f s = f(r)

(r ∈ A)

for some choice of the k variables of r (the active variables). Reordering, if necessary, we may assume that the active variables precede the passive ones. Choose now the set S to be the Cartesian product A × Rn−k , i.e. S = (r, s) r ∈ A and s ∈ Rn−k , and F to be the function

F(r, s) = s − f(r).

CHAPTER 2. MANIFOLD

21

Then S is an open subset8 of Rn and F is continuously differentiable in S. Moreover then F′ = −f ′ In−k

is of full rank (In−k is the (n − k) × (n − k) identity matrix).

Excluding the n-dimensional manifolds, manifolds of Rn are thus exactly all sets which are local loci. In particular conditions of the form G(p) = c

or G(p) − c = 0,

where c is a constant, define a manifold (with the given assumptions), the so-called level manifold of G. Representation of a manifold using loci is often called an implicit representation and the representation using local graphs of functions—as in the original definition– is called an explicit representation.9 Example. 2-dimensional manifolds in R3 are smooth surfaces. Locally such a surface is defined as the set determined by a condition F (x, y, z) = 0, where in the points of interest F′ =

∂F ∂F ∂F 6= 0. , , ∂x ∂y ∂z

In particular surfaces determined by conditions of the form G(x, y, z) = c, where c is constant, are level surfaces of G. For such surfaces it is often quite easy to check whether or not a point p0 = (x0 , y0 , z0 ) is in the surface. Just calculate (locally) F (x0 , y0 , z0 ) and check whether or not it is = 0, of course even this could turn out to be difficult. Example. 1-dimensional manifolds in R3 are smooth space curves. Locally a curve is the locus of points satisfying a condition ( F1 (x, y, z) = 0 F(x, y, z) = 0 or F2 (x, y, z) = 0, where in the curve the derivative matrix F′ =

F1′ F2′

!

is of full rank, i.e., its two rows are linearly independent. Locally we have then the curve as the intersection of the two smooth surfaces F1 (x, y, z) = 0 and F2 (x, y, z) = 0 If B is the r0 -centered ball of radius R in A and s0 ∈ Rn−k , then the (r0 , s0 )-centered open ball of radius R in R is included in S, because in this ball

2 kr − r0 k2 ≤ kr − r0 k2 + ks − s0 k2 = (r, s) − (r0 , s0 ) < R, 8

n

so that r ∈ B. 9 There is a third representation, so-called parametric representation, see Section 2.5.

CHAPTER 2. MANIFOLD

22

(locus conditions, cf. the previous example). It may be noted that the curve of intersection of two smooth surfaces need not be a smooth manifold (curve), for this the full rank property is needed.10 Example. The condition F (x, y) = yey − x = 0

defines a 1-dimensional manifold (a smooth curve, actually a graph) of R2 . On the other hand, F ′ (x, y) = − 1, (1 + y)ey , so, except for the point (−1/e, −1), the corresponding local graph can be taken as f (y), y where f is one of the so-called Lambert W functions11 W0 (green upper branch) or W−1 (red lower branch), see the figure below (Maple). Since here √ y = xe−y and − ln 2 > −1/e, this means that √

√

2

√

2

2

√ ·.· 2

exists, which may seem odd because Actually it has the value √ W0 − ln 2 √ = 2. − ln 2

√

2 > 1.

2.4 Mapping Manifolds. Coordinate-Freeness Manifolds are often ”manipulated” by mapping them by some functions in one way or another. For manifolds defined as in the previous sections it then is not usually at all easy to show that the resulting set really is a manifold. For parametrized manifolds this is frequently easier, see the next section. On the other hand, inverse images come often just as handy: Theorem 2.5. If • M is a k-dimensional manifold of Rn included in the open set B, • A is an open subset of Rm , where m ≥ n, and • g : A → B is a continuously differentiable function having a derivative matrix g′ of full rank (i.e. linearly independent rows), then the inverse image g−1 (M) is an m − n + k-dimensional manifold of Rm . 10

For instance, the intersection of the surface F1 (x, y, z) = z − xy = 0 (a saddle surface) and the surface F2 (x, y, z) = z = 0 (the xy-plane) is not a manifold (and why not?). 11 Very useful in many cases.

CHAPTER 2. MANIFOLD

23

Proof. The case k = n is clear. Then M is an open subset of Rn and its inverse image g−1 (M) is an open subset of Rm , i.e. an m-dimensional manifold of Rm . Take then the case k < n. Consider an arbitrary point r0 of g−1(M), i.e. a point such that g(r0 ) ∈ M. Locally near the point p0 = g(r0 ) the manifold M can be determined as a locus, by Theorem 2.4. More specifically, there is an open subset S of Rn and a function F : S → Rn−k , such that the conditions 1.– 4. of the Implicit Function Theorem are satisfied. For a continuous function defined in an open set the inverse image of an open set is open, so g−1 (S) is open. The locus condition F g(r) = 0 determines locally some part of the set g−1 (M). In the open set g−1 (S) the composite function F ◦ g now satisfies the conditions 1.– 4. of the Implicit Function Theorem since (via chain rule) its derivative is (F ◦ g)′ (r0 ) = F′ g(r0 ) g′ (r0 ) = F′ (p0 )g′ (r0 ), and it is of full rank. Thus, by Corollary 2.3, g−1 (M) is a manifold of Rm and its dimension is m − (n − k) (the dimension of r minus the dimension of F).

So far we have not considered coordinate-freeness of manifolds. A manifold is always expressly defined in some coordinate system, and we can move frome one system to another using coordinate transforms. But is a manifold in one coordinate system also a manifold in any other, and does the dimension then remain the same? As a consequence of Theorem 2.5 the answer is positive. To see this, take a coordinate transform r∗ = rQ + b, and choose m = n and g(r∗ ) = (r∗ − b)QT in the theorem. Then a manifold M in coordinates r∗ is the inverse image of the manifold in coordinates r, and thus truly a manifold. Dimension is preserved as well. Being a manifold in one coordinate system guarantees being a manifold in any other coordinates. ”Manifoldness” is a coordinate-free property.

2.5 Parametrized Manifolds Whenever a manifold can be parametrized it will be in many ways easier to handle.12 For instance, defining and dealing with integrals over manifolds then becomes considerably simpler. A parametrization13 of a k-dimensional manifold M of Rn consists of an open subset U of Rk (the so-called parameter domain), and a continuously differentiable bijective function γ:U →M having a derivative matrix γ ′ of full rank. Since the derivative γ ′ is an n×k matrix and k ≤ n, it is the columns that are linearly independent. These columns are usually interpreted as vectors. (It is naturally assumed here that k > 0.) 12

In many textbooks manifolds are indeed defined using local parametrization, see e.g. S PIVAK or O’N EILL . This usually requires the so-called transition rules to make sure that the chart functions are coherent. Our definition, too, is a kind of local parametrization, but not a general one, and not requiring any transition rules. 13 This is often called a smooth parametrization.

CHAPTER 2. MANIFOLD

24

Evidently, if a manifold is the graph of some function, i.e. s = f(r)

(r ∈ A),

it is parametrized, we just take γ(r) = r, f(r) .

U = A and

Also an n-dimensional manifold of Rn , i.e. an open subset A, is parametrized in a natural way, we take A itself as the parameter domain and the identity function as the function γ.

Example. A circle Y : x2 + y 2 = R2 is a 1-dimensional manifold of R2 which cannot be parametrized. To show this, we assume the contrary, i.e., that Y actually can be parametrized, and derive a contradiction. The parameter domain U is then an open subset of the real line, that is, it consists of disjoint open intervals. Consider one of these intervals, say (a, b) (where we may have a = −∞ and/or b = ∞). Now, when a point u in the interval (a, b) moves to the left towards a, the corresponding point γ(u) in the circle moves along the circumference in eiher direction. It cannot stop or turn back because γ is a bijection and γ ′ has full rank. Thus also the limiting point p = lim γ(u) u→a+

is in the circle Y. But we cannot have in U a point v such that p = γ(v). Such a point would be in one of the open intervals in U, and—as above—we see that a sufficiently short open interval (v−ǫ, v+ǫ) is mapped to an open arc of Y containing the point p. This would mean that γ cannot be bijective, a contradiction. On the other hand, if we remove one of the points in Y, say (R, 0), it still remains a manifold (why?) and can be parametrized using the familiar polar angle φ: (x, y) = γ(φ) = (R cos φ, R sin φ) (0 < φ < 2π). Now γ(φ) = (R cos φ, R sin φ) is a continuously differentiable bijection, and γ ′ (φ) =

−R sin φ R cos φ

!

is always 6= 0. Also the inverse parametrization is easily obtained: φ = atan(x, y), where atan is the bivariate arctangent, i.e., an arc tangent giving correctly the quadrant and also values in the coordinate axes, that is  y  arctan for x > 0 and y ≥ 0   x     y   2π + arctan for x > 0 and y < 0   x    y atan(x, y) = π + arctan for x < 0 x    π    for x = 0 and y > 0   2       3π for x = 0 and y < 0. 2

CHAPTER 2. MANIFOLD

25

It can be found, in one form or in another, in just about all mathematical programs. atan is actually continuous and also continuously differentiable—excluding the negative x-axis, see the figure on the right (by Maple)—since (verify!)

10 8 6

∂atan(x, y) y =− 2 ∂x x + y2 and

4 –2

x ∂atan(x, y) = 2 . ∂y x + y2

y=0,

–1

–1 0

Example. A sphere x2 + y 2 + z 2 = R2 cannot be parametrized either. However, if, say, the half great circle x2 + z 2 = R2 ,

–2

2

y 1 2

1x 2

x≥0

is removed then a manifold is obtained which can be parametrized by the familiar spherical coordinates as (x, y, z) = γ(θ, φ) = (R sin θ cos φ, R sin θ sin φ, R cos θ) and the parameter domain is the open rectangle U : 0 < θ < π , 0 < φ < 2π. The derivative matrix 

 R cos θ cos φ −R sin θ sin φ γ ′ (θ, φ) =  R cos θ sin φ R sin θ cos φ  −R sin θ 0

is of full rank, and the inverse parametrization is again easy to find:  θ = arccos z R φ = atan(x, y).

Example. Parametrizarion of a general smooth space curve is of the form r = γ(u) (u ∈ U),

where U is an open interval. (The parameter domain might actually consist of several open intervals but then the curve can be divided similarly.) Here γ is continuously differentiable and γ ′ 6= 0, which guarantees that the curve has a tangent everywhere. Example. Parametrization of a general smooth surface is of the form r = γ(u)

(u ∈ U),

where U is an open subset of R2 . Here γ is continuously differentiable and γ ′ has full rank, i.e., its two columns are linearly independent. This guarantees that the surface has everywhere a normal (the cross product of the two columns of γ ′ (u)).

CHAPTER 2. MANIFOLD

26

Parametrization is at the same time more restrictive and more extensive than our earlier definitions of manifolds: Not all manifolds can be parametrized and not all parametrizations define manifolds. On the other hand, as noted, parametrization makes it easier to deal with manifolds. In integration restrictions of parametrizations can be mostly neglected since they do not affect the values of the integrals, as we will see later. Let us note, however, that if a set is parametrized then at least it is a manifold in a certain localized fashion: Theorem 2.6. If M ⊆ Rn , U is an open subset of Rk , u0 ∈ U, and there is a continuously differentiable bijective function γ : U → M with a derivative γ ′ of full rank, then there is an open subset V of U such that u0 ∈ V and γ(V) is a k-dimensional manifold of Rn . Proof. Consider a point p0 = γ(u0 ) of M. Then the columns of γ ′ (u0 ) are linearly independent, and thus some k rows are linearly independent, too. Reordering, if necessary, we may assume that these rows are the first k rows of γ ′ (u0 ). Let us first consider the case k < n. For a general point p = (r, s) of M, r contains the k first components. We denote further by γ 1 the function consisting of the first k components of γ, and r0 = γ 1 (u0 ). Similarly, taking the last n − k components of γ we get the function γ 2 . The function F(u, r) = r − γ 1 (u)

defined in the open set S = U × Rk (cf. the proof of Theorem 2.4) then satisfies the conditions 1.– 4. of the Implicit Function Theorem. Thus there is a continuously differentiable function g : B → Rk , defined in an open set B, whose graph u = g(r) is included in S, such that r = γ 1 g(r) .

The chart function f in the point p0 is then obtained by taking f(r) = γ 2 g(r) .

Finally we choose V = γ −1 1 (B), an open set. (And where, if anywhere, do we need bijectivity of γ?) The case k = n is similar. The function F(u, p) = p − γ(u) defined in the open set S = U × Rn then satisfies conditions 1.– 4. of the Implicit Function Theorem. Hence there is an open set B, containing the point p0 , and a continuously differentiable function g : B → Rn , whose graph u = g(p) is included in S, such that p = γ g(p) . Thus B ⊆ M. Again we choose V = γ −1 (B), an open set. Parametrization of a manifold M by γ:U →M may be exchanged, a so-called reparametrization, as follows. Take a new parameter domain V ⊆ Rk , and a continuously differentiable bijective function η : V → U,

CHAPTER 2. MANIFOLD

27

such that the derivative η ′ is nonsingular. The new parametrization is then by the composite function γ ◦ η, that is, as r = γ η(v) (v ∈ V). This really is a parametrization since, by the chain rule, γ ◦ η is continuously differentiable and its derivative γ ′ η(v) η ′ (v)

is of full rank. n-dimensional manifolds of Rn , i.e. open subsets, often benefit from reparametrization.

Example. 3-dimensional manifolds of R3 , i.e. open subsets or ’solids’, are often given using parametrizations other than the trivial one by the identity function. Familiar parametrizations of this type are those using cylindrical or spherical coordinates. For instance, the slice of a ball below (an open set) can be parametrized by spherical coordinates as r = (x, y, z) = γ(ρ, θ, φ) = (ρ sin θ cos φ, ρ sin θ sin φ, ρ cos θ), where the parameter domain is the open rectangular prism V : 0 < ρ < R , 0 < θ < π , 0 < φ < α.

z

φ= 0

φ parameter domain

φ= α α y

ρ= R

U π ρ

θ

R

x

Different parametrizations of a manifold may come separately, without any explicit reparametrizations. Even then in principle, reparametrizations are there. Theorem 2.7. Different parametrizations of a manifold can always be obtained from each other by reparametrizations. Proof. Consider a situation where the k-dimensional manifold M of Rn has the parametrizations r = γ 1 (u) (u ∈ U) and r = γ 2 (v) (v ∈ V). An obvious candidate for the reparametrization is the one using η = γ −1 1 ◦ γ 2 , as u = γ −1 γ 2 (v) . 1

This function η is bijective, we just must show that it is continuously differentiable. For this let us first define G(u, v) = γ 1 (u) − γ 2 (v).

CHAPTER 2. MANIFOLD

28

Then the columns of the derivative G′ corresponding to the variable u, i.e. γ ′1 , are linearly independent. Consider then a point r0 = γ 1 (u0 ) = γ 2 (v0 ) of M. Since the k columns of γ ′1 (u0 ) are linearly independent, then some k rows of γ ′1 (u0 ) are also linearly independent. Picking from G the corresponding k components we get the function F. In the open set S = U × V the function F satisfies the conditions 1.– 4. of the Implicit Function Theorem, and then the obtained function f clearly is η. Since the point v0 was an arbitrary point of V, η is continuously differentiable. On the other hand, η ′ is also nonsingular because γ 2 = γ 1 ◦ η, and by the chain rule γ ′2 (v) = γ ′1 η(v) η ′ (v). If now η ′ would be singular in some point of V, then γ ′2 would not have full rank there.

A parametrized manifold may be localized also in the parameter domain: Take an open subset U ′ of U and interprete it as a new parameter domain. The thus parametrized set is again a manifold, and it has a parameter representation (cf. Theorem 2.6). This in fact also leads to a generalization of manifolds. Just parametrize a set N as above using a parameter domain U and a function γ : U → N which is continuously differentiable and whose derivative γ ′ is of full rank, but do not require that γ is bijective. If now for each point p of N there is an open subset Up of U such that • p = γ(u) for some point u of Up , and • γ is bijective when restricted into Up , then as in Theorem 2.6, the parametrization defines a manifold when restricted into Up . The set N itself then need not be a manifold. Generalized manifolds of this kind are called trajectory manifolds. A trajectory manifold can reparametreized exactly as the usual manifold. Example. The subset of R2 parametrized by the polar angle φ given by (x, y) = γ(φ) = r(φ) cos φ, r(φ) sin φ (0 < φ < 2π),

where

φ , 12 is a complicated plane curve, but not a manifold since it passes through the origin six times, see the left figure below (Maple). It is however a 1-dimensional trajectory manifold. The figure on the right is the hodograph (x, y) = γ ′ (φ)T (0 < φ < 2π). r(φ) = ecos φ − 2 cos 4φ + sin5

It shows that γ ′ is of full rank (the curve does not pass through the origin). It also indicates the parameter value φ = 0 (or φ = 2π) could have been included locally. This would not destroy smoothness of the curve. This is common in polar parametrizations. It should also be remembered that even though atan is discontinuous, sin atan(x, y) and cos atan(x, y) are continuously differentiable. So the parameter interval could have been, say, 0 < φ < 4π containg the parameter value φ = 2π. Note also that the polar parametrization allows even negative values of the radius!

CHAPTER 2. MANIFOLD

29

3 6

2

4

2

1 –6

–4

–2

2

4

6

0

–1

0

1

2

3 –2

–1 –4

–2

–6

–8

–3

Many more self-intersections appear when the curve is drawn for the ”full” parameter interval 0 < φ < 24π, outside of which it starts to repeat itself:

CHAPTER 2. MANIFOLD

30

2.6 Tangent Spaces Locally, near a point p0 of Rn , a k-dimensional manifold M is a graph of some k-variate function f, i.e. s = f(r) (r ∈ A) in Rn , and in particular

p0 = r0 , f(r0 ) .

Geometrically the tangent space of M in the point p0 consists of all tangent vectors touching M in p0 . The dimensions k = n and k = 0 are dealt with separately. In the former the tangent space consists of all vectors, in the latter of only the zero vector. In the sequel we assume that 0 < k < n. Locally, near the point r0 , f comes close to its affine approximation, i.e. f(r) ∼ = f(r0 ) + (r − r0 )f ′ (r0 )T . The affine approximation of a function in a point gives correctly the values of the function and its derivative in this point. Let us denote g(r) = f(r0 ) + (r − r0 )f ′ (r0 )T (whence g′ (r0 ) = f ′ (r0 )). Then s = g(r) is a graph which locally touches the manifold M in the point p0 . Geometrically this graph is part of a k-dimensional hyperplane, or a plane or a line in lower dimensions. The tangent space of M in the point p0 , denoted by Tp0 (M), consists of all (tangent) vectors such that the initial point of their representative directed line segments is p0 and the terminal point is in the graph s = g(r), i.e., it consists of exactly all vectors T T r, g(r) − r0 , f(r0 ) = r − r0 , (r − r0 )f ′ (r0 )T ! ! (r − r0 )T Ik = (r − r0 )T , = ′ T ′ f (r0 )(r − r0 ) f (r0 ) where r ∈ Rk and Ik is the k × k identity matrix. In particular r = r0 is included, so the zero vector always is in a tangent space. In a sense the above tangent space is thus the graph of the vector-valued function T(h) = f ′ (r0 )h of the vector variable h. Apparently T is a linear function and f ′ (r0 ) is the corresponding matrix. Note that replacing the graph s = f(r) by another graph t = h(u) (as needs to be done when moving from one chart to another) simply corresponds to a local reparametrization u = η(r) and change of basis of the tangent space using the matrix η ′ (r0 ) (cf. Theorem 2.7 and its proof). The space itself remains the same, of course. Example. A smooth space curve or a 1-dimensional manifold MÊof R3 is locally a graph (y, z) = f(x) = f1 (x), f2 (x)

(or one of the other two alternatives). The tangent space of M in the point p0 = (x0 , y0, z0 ), where (y0 , z0 ) = f(x0 ), consists of exactly all vectors

CHAPTER 2. MANIFOLD  h  f ′ (x0 )h    1 

f2′ (x0 )h

31

z (h ∈ R).

p0 = (x0 , f1(x0) , f2(x0))

Geometrically the vectors are directed along the line r = p0 + tv (t ∈ R), where v = 1, f1′ (x0 ), f2′ (x0 ) .

y space curve + tangent vector

x

Example. A smooth surface in R3 is a 2-dimensional manifold M. Locally M is the graph z = f (x, y) (or then one of the other two alternatives). The tangent space of M in the point p0 = (x0 , y0, z0 ), where z0 = f (x0 , y0 ), consists of exactly all vectors   1 0   h1 0 1   ((h1 , h2 ) ∈ R2 ).  ∂f (x0 , y0 ) ∂f (x0 , y0 )  h2 ∂x ∂y Geometrically the vectors are thus in the plane r = p0 + t1 v1 + t2 v2

z

(t1 , t2 ∈ R),

v1

tangent plane

where

and

∂f (x0 , y0 ) v1 = 1, 0, ∂x ∂f (x0 , y0) . v2 = 0, 1, ∂y

p0 = (x0 , y0 , f(x0,y0)) y v2 x

What about when a manifold M is given by local loci, say locally by the condition F(r, s) = 0 (assuming a proper order of variables)? According to Corollary 2.3, then M is given locally near the point p0 = (r0 , s0 ) also as a graph s = f(r) and (cf. the Implicit Function Theorem) −1 f ′ (r0 ) = −F′s r0 , f(r0 ) F′r r0 , f(r0 ) where

F′ = F′r F′s .

The tangent space Tp0 (M) consists of the vectors ! Ik f ′ (r0 )

h.

But these are exactly all vectors m=

h k

CHAPTER 2. MANIFOLD

32

satisfying the condition

i.e.

F′s r0 , f(r0 ) k + F′r r0 , f(r0 ) h = 0, F′ (p0 )m = 0.

So we get Theorem 2.8. If a manifold M is locally near the point p0 given as the locus defined by the condition F(p) = 0 (with the assumptions of Corollary 2.3), then the tangent space Tp0 (M) is the null space of the matrix F′ (p0 ). In practice it of course suffices to find a basis for the tangent space (or null space). Coordinate-freeness of tangent spaces has not been verified yet, but as a consequence of the theorem, Corollary 2.9. Tangent spaces of manifolds are coordinate-free. Proof. This follows because a null space is coordinate-free, and a manifold can be given as a local locus (Theorem 2.4). More specifically, if we take a coordinate transform p∗ = pQ + b and denote F∗ (p∗ ) = F (p∗ − b)QT and m∗ = QT m, then (cf. Section 1.5)

F∗ ′ (p∗0 )m∗ = F′ (p∗0 − b QT )QQT m = F′ (p0 )m.

Moreover, 0-dimensional and n-dimensional manifolds of Rn (points and open sets) of course are coordinate-free. Example. The tangent space of a circle F (x, y) = x2 + y 2 − R2 = 0 in the point (x0 , y0) is then the null space of the 1 × 2 matrix F ′ (x0 , y0 ) = (2x0 , 2y0). h satisfying It consists of vectors k 2x0 h + 2y0 k = 0 (cf. the equation of a line). Similarly the tangent space of a sphere F (x, y, z) = x2 + y 2 + z 2 − R2 = 0 in the point (x0 , y0, z0 ) is the null space of the 1 × 3 matrix F ′ (x0 , y0 , z0 ) = (2x0 , 2y0 , 2z0 ).   h  It consists of vectors k  satisfying l 2x0 h + 2y0 k + 2z0 l = 0 (cf. the equation of a plane). Tangent spaces of second degree curves and surfaces are sometimes called polar spaces.

CHAPTER 2. MANIFOLD

33

Example. In general, the tangent space of a smooth surface (manifold) M in R3 defined implicitly by the equation F (x, y, z) = 0 in the point p0 = (x0 , y0 , z0 ) is the null space of the 1 × 3 matrix F ′ (p0 ), i.e., the set of vectors m = (h, k, l)T satisfying F ′ (p0 ) • m =

∂F (p0 ) ∂F (p0 ) ∂F (p0 ) h+ k+ l = 0. ∂x ∂y ∂z

As a further consequence of Theorem 2.8 we see that if a manifold is parametrized, then its tangent space can be parametrized, too. Corollary 2.10. If the k-dimensional manifold M of Rn has the parametrization γ : U → M and p0 = γ(u0 ), then the tangent space Tp0 (M) consists of exactly all vectors of the form γ ′ (u0 )v

(v ∈ Rk ),

that is, Tp0 (M) is the column space of γ ′ (u0 ). Proof. Locally near the point p0 the manifold M can be given as a locus determined by some suitable condition F(p) = 0. Thus the equation F γ(u) = 0

is an identity valid in a neighborhood of the parameter value u0 . Applying the chain rule we get another identity F′ (γ u) γ ′ (u) = O, where O is a zero matrix of appropriate dimensions. Substituting u = u0 we get the equation F′ (p0 )γ ′ (u0 ) = O,

showing that columns of γ ′ (u0 ) are in the null space of F′ (p0 ). On the other hand, since the dimension of the null space is k and the k columns of γ ′ (u0 ) are linearly independent, the columns of γ ′ (u0 ) span the tangent space Tp0 (M). Example. If the parametrization of a smooth space curve C (a 1-dimensional manifold of R3 ) is r = γ(u) (u ∈ U),

then its tangent space Tr0 (C) in the point r0 = γ(u0 ) consists of exactly all vectors hγ ′ (u0 )

(h ∈ R).

Example. If the parametrization of a smooth surface of R3 (a 2-dimensional manifold) is r = γ(u)

(u ∈ U),

then its tangent space in the point r0 = γ(u0 ) consists of exactly all vectors   ∂γ1 (u0 ) ∂γ1 (u0 )  ∂u1 ∂u2      ∂γ(u0 ) ∂γ(u0 ) ∂γ (u ) ∂γ (u )  2 0  h1 2 0 ′ h = γ (u0 )h =    ∂u1 ∂u2  h2 ∂u1 ∂u2    ∂γ3 (u0 ) ∂γ3 (u0 )  ∂u1 ∂u2

(h ∈ R2 ).

CHAPTER 2. MANIFOLD

34

2.7 Normal Spaces Geometrically the normal space of a k-dimensional manifold M of Rn in the point p0 , denoted Np0 (M), consists of exactly all (tangent) vectors perpendicular to to all vectors of the tangent space. In other words, the normal space is the orthogonal complement of the tangent space. Again the cases k = n and k = 0 are special and are omitted in the sequel. In the former the normal space consists of the the zero vector only, and in the latter of all vectors. Vectors in a normal space are called normals or normal vectors. Basic properties of a normal space Np0 (M) follow fairly directly from those of the tangent space. We just list them here. From basic courses of mathematics we remember that the null space of a matrix is the orthogonal complement of the column space of its transpose, and that the column space is the orthogonal complement of the null space of its transpose. • If the k-dimensional manifold M of Rn is near the point p0 locally a graph s = f(r), then its normal space Np0 (M) consists of exactly all vectors ′ −f (r0 )T k (k ∈ Rn−k ). In−k • If the k-dimensional manifold M of Rn is near the point p0 locally given as a locus determined by the condition F(p) = 0 (with the assumptions of Corollary 2.3), then its normal space Np0 (M) is the column space of the matrix F′ (p0 )T , i.e., it consists of exactly all vectors F′ (p0 )T k (k ∈ Rn−k ). • If the k-dimensional manifold M of Rn is parametrized by γ : U → M and p0 = γ(u0 ), then the normal space Np0 (M) is the null space of γ ′ (u0 )T , i.e., it consists of exactly all vectors n satisfying γ ′ (u0 )T n = 0. • A normal space is coordinate-free. • The dimension of the a normal space of a k-dimensional manifold of Rn is always n − k.

z

Example. A smooth space curve, i.e. a 1dimensional manifold M of R3 , is locally a graph (y, z) = f(x) = f1 (x), f2 (x) (or one of the other two alternatives). The normal space of M in the point p0 = (x0 , y0 , z0 ), where (y0 , z0 ) = f(x0 ), then consists of exactly all vectors

p0 = (x0 , f1(x0) , f2(x0))

y space curve + normal plane

x

 −f1′ (x0 ) −f2′ (x0 )  k1  1 0 k2 0 1 

(k1 , k2 ∈ R).

Geometrically these vectors are in the plane

(x − x0 ) + f1′ (x0 )(y − y0 ) + f2′ (x0 )(z − z0 ) = 0.

CHAPTER 2. MANIFOLD

35

Some normals are more interesting than others, dealing with curvature, torsion, and the plane most accurately containing the curve near p0 . Example. A smooth surface in R3 is a 2-dimensional manifold M. Locally M is a graph z = f (x, y) (or one of the other two alternatives). The normal space of M in the point p0 = (x0 , y0, z0 ), where then z0 = f (x0 , y0 ), consists of exactly all vectors   ∂f (x0 , y0 ) k − ∂x    ∂f (x , y )  0 0   (k ∈ R). k − ∂y   k Geometrically these vectors are in the line r = p0 + tv (t ∈ R), where ∂f (x , y ) ∂f (x , y ) 0 0 0 0 v= − ,− ,1 . ∂x ∂y

Example. If the parametrization of a smooth surface (a 2-dimensional manifold of R3 ) is r = γ(u)

(u ∈ U),

then its normal space in the point r0 = γ(u0 ) consists of exactly all vectors   h n = k  l

satisfying

 ∂γ1 (u0 ) ∂γ2 (u0 ) ∂γ3 (u0 )   h  ∂u1 ∂u1 ∂u1  0 ′ T     . k = γ (u0 ) n =   0 ∂γ1 (u0 ) ∂γ2 (u0 ) ∂γ3 (u0 ) l ∂u2 ∂u2 ∂u2 The basis vector of the null space is now obtained in the usual way using cross product, thus the vectors are ∂γ(u ) ∂γ(u ) 0 0 t × (t ∈ R). ∂u1 ∂u2 (The cross product is not the zero vector because the columns of γ ′ (u0 ) are linearly independent.) For instance, the normal space of a sphere parametrized by spherical coordinates as 

(x, y, z) = γ(θ, φ) = (R sin θ cos φ, R sin θ sin φ, R cos θ) (0 < θ < π , 0 < φ < 2π) in the point γ(θ0 , φ0 ) consists of the vectors      2 2  R cos θ0 cos φ0 −R sin θ0 sin φ0 R sin θ0 cos φ0 t  R cos θ0 sin φ0  ×  R sin θ0 cos φ0  = t  R2 sin2 θ0 sin φ0  −R sin θ0 0 R2 sin θ0 cos θ0 i.e. vectors tγ(θ0 , φ0 )T

(t ∈ R).

(t ∈ R),

CHAPTER 2. MANIFOLD

36

2.8 Manifolds and Vector Fields To deal with the spaces we choose bases for the tangent space Tp0 (M) and the normal space Np0 (M) of a k-dimensional manifold M of Rn in the point p0 : t1 , . . . , tk

and n1 , . . . , nn−k ,

respectively. (These bases need not be normalized nor orthogonal.) Note that it was always easy to get a basis for one of the spaces above. Since the two spaces are orthogonal complements of each other, combining the bases gives a basis of Rn . A vector field F may be projected to these spaces in the point p0 : F(p0 ) = Ftan (p0 ) + Fnorm (p0 ). Here Ftan (p0 ) is the flux of the field in the manifold M and Fnorm (p0 ) is the flux of the field through the manifold M in the point p0 . (Cf. flow of a liquid through a surface.) It naturally suffices to have one of these, the other is obtained by subtraction. From the bases we get the matrices T = (t1 , . . . , tk ) and

N = (n1 , . . . , nn−k )

and further the nonsingular matrices (so-called Gramians) G = TT T = (Gij ) , T

H = N N = (Hij ) , Let us further denote

where and

where

where Gij = ti • tj , and

where Hij = ni • nj .

 a1   a = TT F(p0 ) = TT Ftan (p0 ) =  ...  , ak 

ai = F(p0 ) • ti = Ftan (p0 ) • ti , 

 b1   b = NT F(p0 ) = NT Fnorm (p0 ) =  ...  , bn−k bi = F(p0 ) • ni = Fnorm (p0 ) • ni .

Since dot product is coordinate-free, elements of these matrices and vectors are so, too. The components being in their corresponding spaces we can write Ftan (p0 ) = Tc and

Fnorm (p0 ) = Nd

for vectors c and d. Solving these from the equations a = TT Tc and

b = NT Nd

we see that the components of the field are given by the formulas14 Ftan (p0 ) = T(TT T)−1 TT F = TG−1 a and Fnorm (p0 ) = N(NTN)−1 NT F = NH−1 b. 14

The least squares formulas, probably familiar from many basic courses of mathematics. Note that TT T is nonsingular because otherwise there would be a nonzero vector c such that cT TT Tc = 0, i.e. Tc = 0.

CHAPTER 2. MANIFOLD Example. The flux of a vector field in a smooth surface (a 2-dimensional manifold of R3 ) and through it is obtained by projecting the field to a (nonzero) normal vector n. Here N = n , H = n • n = knk2 , b = F(p0 ) • n

37

z

F

normal vector Fnorm tangent plane

p0 Ftan

y

x and (as is familiar from basic courses) n n 1 Fnorm (p0 ) = F(p0 ) • =n F(p ) • n . 0 knk knk knk2 Note. A k-dimensional manifold in Rn together with its k-dimensional tangent spaces may be thought of, at least locally, as a 2k-dimensional manifold (in Rn+k ), the so-called tangent bundle, and the tangent spaces are its fibres. Similarly the manifold together with its n − k-dimensional normal spaces can be thought of as an n-dimensional manifold (in R2n−k ), the normal bundle. The flux of a vector field in the manifold may then be seen locally as a cross-section of the tangent bundle parametrized by the manifold, and similarly the flux through the manifold as a cross-section of the normal bundle. This kind of geometric view of vector fields (and more generally tensor fields) is popular in modern physics, but is pursued no further here. A more general concept is the fiber bundle. See e.g. A BRAHAM & M ARSDEN & R ATIU .

”Calculating surface area is a foolhardy enterprise; fortunately one seldom needs to know the area of a surface. Moreover, there is a simple expression for dA which suffices for theoretical considerations.” (M ICHAEL S PIVAK : Calculus on Manifolds)

Chapter 3 VOLUME 3.1 Volumes of Sets Geometrical things such as the area of a square or the volume of a cube can be defined using lengths of their sides and edges. If the objects are open sets, these are already examples of volumes of manifolds, as is length of an open interval. A square may be situated in threedimensional space, squares having the same length of sides are congruent and they have the same area. In the n-dimensional space the n-dimensional volume of an n-dimensional cube (or n-cube) is similar: If the edge length is h, then the volume is hn . Such a cube may be situated in an even higher-dimensional space and still have the same (n-dimensional) volume. Such volumes may be thought of as geometric primitives. A way to grasp the volume of a bounded set A ⊂ Rn is the so-called Jordan measure. For this we need first two related concepts. An outer cover P of A consists of finitely many similar n-cubes in an n-dimensional grid, the union of which contains A. Adjacent n-cubes share a face but no more or no less. The family of all outer covers is denoted by Pout . An inner cover P consists of finitely many similar n-cubes in a grid, the union of which is contained in A. The family of all inner covers is denoted by Pin . (This Pin may well be empty.) Note that lengths of edges or orientations of the grid are in no way fixed, and neither is any coordinate system. The volume of a cover P , denoted by |P |, is the sum of the volumes its n-cubes. (And this is a geometric primitive.) The volume of the empty cover is = 0. Quite obviously, the volume of any inner cover is at most the volume of every outer cover, outer covers covering all inner covers.

inner cover

outer cover

The Jordan outer and inner measures of the set A are |A|out = inf |P | and P ∈Pout

|A|in = sup |P |, P ∈Pin

respectively. A set A is Jordan measurable if |A|out = |A|in, and the common value |A| is its 38

CHAPTER 3. VOLUME

39

Jordan’s measure1 . This measure is now defined to be the volume of A. Such a volume clearly is coordinate-free. The precise same construct can be used in a k-dimensional space embedded in an n-dimensional space (where n > k). The k-dimensional space is then an affine subspace of Rn , i.e., a manifold R parametrized by γ(u) = b +

k X

ui vi

i=1

(u ∈ Rk ),

where v1 , . . . , vk are linearly independent. Taking b as the origin and, if needed, orthogonalizing v1 , . . . , vk we may embed Rk in an obvious way in Rn , and thus define the k-dimensional volume of a bounded subset of R. (The n-dimensional volume of these subsets in Rn is = 0, as is easily seen.) Such affine subspaces are e.g. planes and lines of R3 . Note. Not all bounded subsets of Rn have a volume! There are bounded sets not having a Jordan measure. The inner and outer covers used in defining Jordan’s inner and outer measures remind us of the n-dimensional Riemann integral, familiar from basic courses of mathematics, with its partitions and lower and upper sums. It is indeed fairly easy to see that whenever the volume exa3 ists, it can be obtained by integrating the constant 1, with improper integrals, if needed: a1 Z |A| = 1 dr. A

An important special case is the volume of a parallelepiped of Rn (of R3 on the right).

a2

Theorem 3.1. The volume of the parallelepiped P in Rn with edges given by the vectors a1 , a2 , . . . , an (interpreted as line segments) is |P| = det(A) , where A = (a1 , a2 , . . . , an ) is the matrix with columns a1 , a2 , . . . , an .

Proof. The case is clear if a1 , a2 , . . . , an are linearly dependent, the volume is then = 0. Let us then consider the more interesting case where a1 , a2 , . . . , an are linearly independent. The volume is given by the integral above, with a change of variables by r = uAT + b. A wellknown formula then gives Z Z |P| = 1 dr = det(A) du = det(A) |C|, P

C

where C is the unit cube in Rn given by 0 ≤ ui ≤ 1 (i = 1, . . . , n). The volume of this cube is = 1. 1

Also called Jordan–Peano measure or Jordan’s content.

CHAPTER 3. VOLUME

40

We already know volumes are coordinate-free. For the parallelepiped this is also clear because the volume can be written as p |P| = det(G),

where G is the Gramian (cf. Section 2.8)

G = AT A = (Gij )

and Gij = ai • aj .

Being dot products, elements of a Gramian are coordinate-free. The same formula is valid when we are dealing with the k-dimensional volume of a k-dimensional parallelepiped P as part of Rk embedded as an affine subspace in Rn , and P is given by the n-dimensional vectors a1 , a2 , . . . , ak . The Gramian is here, too, a k × k matrix. Note that we need no coordinate transforms or orthogonalizations, the volume is simply given by the n-dimensional vectors. An example would be a parallelogram embedded in R3 , with its sides given by 3-dimensional vectors. A bounded subset A of Rn is called a null set if its volume (or Jordan’s measure) is = 0. For a null set A then |A|out = inf |P | = 0 P ∈Pout

(the empty inner cover is always available). An unbounded subset A is a null set if its all (bounded) subsets r r ∈ A and krk ≤ N (N = 1, 2, . . . ) are null sets. Using the above definition it is often quite difficult to show a set is a null set. A helpful general result is Theorem 3.2. If M is a k-dimensional manifold of Rn and k < n, then bounded subsets of M are null sets of Rn . Proof. The proof is based on a tedious and somewhat complicated estimation, see e.g. H UB BARD & H UBBARD . Just about the only easy thing here is that if the volume of a bounded subset A exists, then it is = 0. Otherwise |A|in > 0 and some inner cover of A would have at least one n-dimensional cube contained in M. But near the center of the cube M is locally a graph of a function, which is not possible. Example. Bounded subsets of smooth surfaces and curves of R3 (1- and 2-dimensional manifolds) are null sets. Null sets—as well as the k-null-sets to be defined below—are very important for integration since they can be included and excluded freely in the region of integration without changing values of integrals. Rather than the ”full” n-dimensional one, it is possible to define a lower-dimensional volume for subsets of Rn , but this is fairly complicated in the general case.2 On the other hand, it is easy to define the k-dimensional volume of an n-cube: It is hk if the edge length of the cube is h. This gives us the k-dimensional volume of an outer cover P of a subset A of Rn , and then, via the infimum, the k-dimensional Jordan outer measure. The corresponding inner measure is of course always zero if A does not contain an open set, i.e. its interior A◦ is empty. Thus, at least it is easy to define the k-dimensional zero volume for a subset A of Rn , that is, the k-null-set of Rn : It is a set having a zero k-dimensional Jordan outer measure. Again this definition is not that easy to use. Theorem 3.2 can be generalized, however, the proof then becoming even more complicated (see H UBBARD & H UBBARD): 2

And has to do e.g. with fractals.

CHAPTER 3. VOLUME

41

Theorem 3.3. If M is a k-dimensional manifold of Rn and k < m ≤ n, then bounded subsets of M are m-null-sets of Rn . For instance, bounded subsets of smooth curves of R3 (1-dimensional manifolds) are 2-null-sets of R3 .

3.2

Volumes of Parametrized Manifolds

In general, the k-dimensional volume of a k-dimensional manifold M of Rn is a difficult concept. For a parametrized manifold it is considerably easier. If the parametrization of M is r = γ(u) then

(u ∈ U),

Z q det γ ′ (u)T γ ′ (u) du, |M|k = U

or briefly denoted |M|k =

Z

dr.

M

Such a volume may be infinite. Note that inside the square root there is a Gramian determinant which is coordinate-free. The whole integral then is coordinate-free. Comparing with the k-dimensional volume of a parallelepiped p |P|k = det(AT A)

in the previous section we notice that in a sense we obtain |M|k by ”summing” over the points r = γ(u) of the manifold k-dimensional volumes of parallelepipeds whose edges are given by the vectors ∂γ(u) dui (i = 1, . . . , k). ∂ui Moreover, (as a directed line segment) the vector dui

∂γ(u) ∂ui

approximatively defines the movement of a point in the manifold when the parameter ui changes a bit the other parameters remaining unchanged. Note. In a certain sense definition of the volume of a parametrized manifold then is natural, but we must remember that it is just that, a definition. Though various problems concerning volumes are largely solved for parametrized manifolds—definition, computing, etc.—others remain. There are parametrizations giving a volume for a manifold even when it does not otherwise exist (e.g. as Jordan measure). On the other hand, a manifold having a volume may sometimes be given a parametrization such that the above integral does not exist. Things thus depend on the parametrization. In the sequel we tacitly assume such ”pathological” cases are avoided.

CHAPTER 3. VOLUME

42

It is important to check that the definition above gives the open parallelepiped P the same volume as before. The parallelepiped is a manifold and can be parametrized naturally as r = γ(u) = b +

k X

ui aTi

(U : 0 < u1 , . . . , uk < 1),

i=1

whence ′

γ (u) = A and

|P|k =

Z p

det(AT A) du =

U

p

det(AT A).

Another important fact to verify is freeness of parametrization. Theorem 3.4. The volume of a parametrized manifold does not depend on the parametrization.3 Proof. By Theorem 2.7 different parametrizations can be obtained from each other by reparametrization. When reparametrizing a k-dimensional manifold M originally parametrized by some γ : U → M a new parameter domain V ⊆ Rk is taken and a continuously differentiable bijective function η : V → U having a nonsingular derivative matrix η ′ . The new parametrization then is given by the composite function δ = γ ◦ η as r = γ η(v) = δ(v) (v ∈ V). Via the chain rule

δ ′ (v) = γ ′ η(v) η ′ (v).

In the integral giving |M|k this corresponds to the change of variables u = η(v) and the corresponding transform of the region of integration from U to V. Thus we only need to check4 the form of the new integrand: Z q det γ ′ (u)T γ ′ (u) du |M|k = = = = =

U Z q

V Z q

V Z q

V Z q V

T det γ ′ η(v) γ ′ η(v) det η ′ (v) dv

T 2 det γ ′ η(v) γ ′ η(v) det η ′ (v) dv T det η ′ (v)T det γ ′ η(v) γ ′ η(v) det η ′ (v) dv det δ ′ (v)T δ ′ (v) dv.

Example. The 1-dimensional volume of a smooth parametrized space curve C : r = γ(u) (u ∈ U) is its length since Z p Z Z q

′ T ′ ′ ′ det γ (u) γ (u) du = γ (u) • γ (u) du = γ ′ (u) du. |C|1 = U

3

4

U

U

Remember that we tacitly assume all parametrizations do give a volume. Recall that for square matrices det(AB) = det(A) det(B) and det(AT ) = det(A).

CHAPTER 3. VOLUME

43

Example. Similarly the 2-dimensional volume of a parametrized surface S : r = γ(u) is |S|2 =

Z q U

(u ∈ U)

det γ ′ (u)T γ ′ (u) du.

This is the same as the familiar area Z

∂γ(u) ∂γ(u) ×

du ∂u1 ∂u2 U

since the area (2-dimensional volume) of the parallelogram in the integrand (a 2-dimensional parallelepiped) can be given as the length of the cross product. Note. The k-dimensional volume of a k-dimensional trajectory manifold of Rn is defined analogously, and it, too, does not depend on the parametrization. Of course, the difficulties mentioned in the previous note remain true for trajectory manifolds, too.

3.3 Relaxed Parametrizations Since the k-dimensional volume of k-null-sets of Rn is = 0, adding such sets in a k-dimensional manifold (via union) does not change the k-dimensional volume of the manifold. When defining a parametrized manifold as in Section 2.5 often parts of the manifold must be removed first. For instance, parametrizing a sphere using spherical coordinates does not work as such, but only after parts of the surface are removed (e.g. a half great circle). Properly extending parametrization these excluded parts may be included again, at least as far as volumes and other integrations are concerned. A relaxed parametrization5 of a k-dimensional manifold M of Rn is obtained by spesifying an extended parameter domain U ⊆ Rk , an exception set X ⊂ U, and a continuous function γ : U → Rn , such that 1. M ⊆ γ(U) and γ(U − X ) ⊆ M (frequently M = γ(U)). 2. The boundary ∂U of U is a null set of Rk .6 Often ∂U is at least partly included in the exception set X . 3. X is a null set of Rk . 4. γ(X ) is a k-null-set of Rn . 5. The set M′ = γ(U − X ) is a k-dimensional manifold of Rn parametrized by r = γ(u)

(u ∈ U − X )

(in the sense of Section 2.5). 5

This is in no way a standard concept in the literature, where several kinds of relaxed parametrizations appear. Here the concept is so general that according to H UBBARD & H UBBARD every manifold has an relaxed parametrization. 6 This strange-looking condition is needed to exclude certain ”pathological” situations. Open subsets of Rk may actually have a boundary that is not a null set.

CHAPTER 3. VOLUME

44

Note that item 5. implies γ is continuously differentiable and bijective in U − X . In U it is just continuous and not necessarily bijective. Furthermore, U − X is an open set but U might not be. Thus, should U contain points of its boundary ∂U, they must be in X , too.

φ

Example. Parametrization of a sphere by spherical coordinates in a form where the parametrization is extended to the whole sphere is an example of a relaxed parametrization:

2π

r = γ(θ, φ) = (R sin θ cos φ, R sin θ sin φ, R cos θ)

X= U

U

(U : 0 ≤ θ ≤ π , 0 ≤ φ ≤ 2π). The exception set X is here the boundary ∂U of U. The corresponding ball (a 3-dimensional manifold of R3 ) in turn is obtained by the relaxed parametrization

π

θ

r = γ(ρ, θ, φ) = (ρ sin θ cos φ, ρ sin θ sin φ, ρ cos θ) (U : 0 ≤ ρ ≤ R , 0 ≤ θ ≤ π , 0 ≤ φ ≤ 2π). U is a rectangular prism whose boundary is the exception set. Example. After removing the four vertices, the perimeter of a square is a 1-dimensional manifold M of R2 , consisting of four separate parts, with the relaxed parametrization   (u + 3, −1) for −4 ≤ u ≤ −2     (1, u + 1) for −2 ≤ u ≤ 0 γ(u) =  (1 − u, 1) for 0 ≤ u ≤ 2      (−1, 3 − u) for 2 ≤ u ≤ 4,

y 1

M

1

1

x

1

where the parameter domain is U : −4 ≤ u ≤ 4. The exception set is X = {−4, −2, 0, 2, 4}. Similarly, relaxed parametrizations could be obtained for surfaces of cubes, or more generally, surfaces of polyhedra. In volume computations the exception set X has no contribution since it is mapped to γ(X ), a k-null-set of Rn : Z q ′ det γ ′ (u)T γ ′ (u) du. |M|k = |M |k = U −X

q

If the integrand det γ ′ (u)T γ ′ (u) has a continuous extension onto the whole U, as is sometimes the case, we can write further Z q det γ ′ (u)T γ ′ (u) du, |M|k = U

since X is a null set of Rk . Improper integrals are here allowed, too, giving even more ”relaxation”. Thus relaxed parametrizations can be used in volume computations and integrations more or less as the ”usual” parametrizations of Section 2.5.

CHAPTER 3. VOLUME

45

Even relaxed reparametrization7 is possible and preserves volumes. A new extended parameter domain V ⊆ Rk , an exception set Y ⊂ V, and a continuous function η : V → U are then sought, such that 1. restricted to Y, η is a bijective function Y → X . 2. restricted to V − Y, η is a continuously differentiable bijective function V − Y → U − X , and its derivative η ′ is nonsingular (i.e. η then gives the ”usual” reparametrization). In such a relaxed reparametrization the exception set is mapped to an exception set, which guarantees preservation of the volume in reparametrization (as a consequence of Theorem 3.4). To make reparametrization in this sense possible, exception sets often need to be modified. Example. The circle x2 + y 2 = 1 is a 1-dimensional manifold of R2 having e.g. the following two relaxed parametrizations: r = γ 1 (φ) = (cos φ, sin φ)

(0 ≤ φ < 2π)

(the usual polar coordinate parametrization with the exception set {0}), and  p  − u − 2, 1 − (u + 2)2 for −3 ≤ u ≤ −1 r = γ 2 (u) = (U : −3 ≤ u < 1)  u, −√1 − u2 for −1 ≤ u < 1

(the exception set is {−3, −1}). These two relaxed parametrizations cannot be directly reparametrized to each other. If, however, in the exception set of the first parametrization the ”unnecessary” number π is added, then a reparametrization is possible by the function ( −2 − cos φ for 0 ≤ φ ≤ π u = η(φ) = cos φ for π ≤ φ < 2π. Note. Relaxed parametrizations are possible for trajectory manifolds, too. And volumes will then also be preserved in relaxed reparametrizations.

7

Again in literature this is defined in many different ways.

”Gradient a 1-form? How so? Hasn’t one always known the gradient as a vector? Yes, indeed, but only because one was not familiar with the more appropriate 1-form concept. The more familiar gradient is the vector corresponding to the 1-form gradient.” (C.W. M ISNER & K.S. T HORNE & J.A. W HEELER : Gravitation)

Chapter 4 FORMS 4.1 k-Forms Whenever vector fields are integrated—the result being a scalar—the fields first need to be ”prepared” in one way or another, so that the integrand is scalar-valued. A pertinent property of a vector field is its direction, which then must be included somehow in the preparation. Integration regions are here parametrized manifolds, possibly in a relaxed sense. Directions related to the manifold can be included via tangent spaces or normal spaces. And in many cases orientation of the manifold should be fixed, too. Examples of integrals of this type are the familiar line and surface integrals Z Z F(r) • ds and F(r) • dS. C

S

A general apparatus for the ”preparation” is given by so-called forms. In integration they appear formally as a kind of differentials, and are therefore often called differential forms. In various notations, too, differentials appear for this reason. Other than this, they do not have any real connection with differentials. An n-dimensional k-form is a function φ mapping n-dimensional vectors1 r1 , . . . , rk (thus k vectors, and the order is relevant!) to a real number φ(r1 , . . . , rk ), satisfying the conditions 1. φ is antisymmetric, i.e., interchanging any two vectors ri and rj changes the sign of the value φ(r1 , . . . , rk ). Thus if for instance k = 4, then φ(r3 , r2 , r1 , r4 ) = −φ(r1 , r2 , r3 , r4 ). 2. φ is multilinear, i.e., it is linear with respect to any argument position (vector ri ): φ(r1 , . . . , ri−1 , c1 ri1 + c2 ri2 , ri+1 , . . . , rk ) = c1 φ(r1 , . . . , ri−1 , ri1 , ri+1, . . . , rk ) + c2 φ(r1 , . . . , ri−1 , ri2 , ri+1, . . . , rk ) (similarly the cases i = 1 and i = n). As a limiting case, 0-forms can be included, too: they are constant functions which do not have any variable vectors at all. 1

Often tangent vectors where only the vector part is used. Note that in 1-, 2- and 3-dimensional spaces geometric vectors could be used.

46

CHAPTER 4. FORMS

47

Thinking about the intended use of k-forms—as devices for coupling vector fields and k-dimensional volumes for integration—this definition looks fairly minimal. Antisymmetry makes it possible the change direction by ”reflecting”, that is, interchanging two vector variables. Multilinearity on the one hand guarantees the volume of a combination of two disjoint sets is duly obtained by adding the volumes of the two parts, and on the other hand implies that scaling works correctly, i.e., if the vector variables are multiplied (scaled) by the scalar λ, then the value is scaled by λk . Example. A familiar example of an n-dimensional n-form is the determinant det. Take n vectors r1 , . . . , rn , consider them as columns of a matrix: (r1 , . . . , rn ), and compute its determinant det(r1 , . . . , rn ). It follows directly from basic properties of determinants that this is an n-form. Example. An equally familiar example of an n-dimensional 1-form is dot product with a fixed vector. If a is a fixed constant vector, then the function φ(r) = a • r is a 1-form. Note that antisymmetry is not relevant here because there is only on vector variable. The dot product r1 • r2 is however not a 2-form (and why not?). Example. A further familiar example of a 3-dimensional 2-form is the first component (or any other component) of the cross product r1 × r2 . Properties of the cross product immediately imply that (r2 × r1 )1 = (−r1 × r2 )1 = −(r1 × r2 )1 (antisymmetry) and that r1 × (c1 r21 + c2 r22 ) (multilinearity).

1

= (c1 r1 × r21 + c2 r1 × r22 )1 = c1 (r1 × r21 )1 + c2 (r1 × r22 )1

Let us list some basic properties of forms: Theorem 4.1. (i) If φ(r1 , . . . , rk ) is an n-dimensional k-form and c is a constant, then cφ(r1 , . . . , rk ) is also a k-form. (ii) If φ1 (r1 , . . . , rk ) and φ2 (r1 , . . . , rk ) are n-dimensional k-forms, then so is their sum φ1 (r1 , . . . , rk ) + φ2 (r1 , . . . , rk ). (iii) If φ(r1 , . . . , rk ) is an n-dimensional k-form and c is a constant vector, then φ1 (r1 , . . . , rk−1) = φ(r1 , . . . , rk−1, c) is an n-dimensional k −1-form. (And similarly when c is in any other argumant position.) (iv) If the vectors a1 , . . . , ak are linearly dependent, then the value of a k-form φ(a1 , . . . , ak ) is = 0. In particular, if some ai is the zero vector, then φ(a1 , . . . , ak ) = 0. Proof. Items (i), (ii) and (iii) are immediate, so let us proceed to item (iv). We notice first that if two of the vectors a1 , . . . , ak are the same, then the value is = 0. Indeed, interchanging these vectors the value changes its sign, and also remains the same. If then a1 , . . . , ak are linearly dependent, then one of them can be expressed as a linear combination of the others, say, ak =

k−1 X i=1

ci ai .

CHAPTER 4. FORMS

48

But then by the multilinearity φ(a1 , . . . , ak ) =

k−1 X

ci φ(a1 , . . . , ak−1 , ai ),

i=1

which is = 0. In particular, if say a1 = 0, then φ(0, a2 , . . . , ak ) = φ(0 · 0, a2 , . . . , ak ) = 0φ(0, a2 , . . . , ak ) = 0. As a consequence of item (iv), an n-dimensional k-form where k > n is rather uninteresting: it always has the value 0. Forms in fact are a kind of generalizations of determinants. To see this, let us see how an n-dimensional k-form φ(r1 , . . . , rk ) is represented when the vectors r1 , . . . , rk are represented in an orthonormalized basis e1 , . . . , en as ri =

n X

xj,i ej

(i = 1, . . . , k).

j=1

Multilinearity of the last argument position rk gives φ(r1 , . . . , rk ) =

n X

xj,k φ(r1 , . . . , rk−1, ej ).

j=1

Continuing using the representation of rk−1 in the basis gives further φ(r1 , . . . , rk ) =

n n X X

xl,k−1 xj,k φ(r1 , . . . , rk−2, el , ej ).

j=1 l=1

Note that all terms having i = j can be omitted in the sum. In this way we finally get a representation of φ(r1 , . . . , rk ) as a sum of terms of the form xj1 ,1 xj2 ,2 · · · xjk ,k φ(ej1 , ej2 , . . . , ejk ) where j1 , j2 , . . . , jk are disjoint indices. Let us then collect together terms where the indices j1 , j2 , . . . , jk are the same, just possibly in a different order. As an example take the case where the indices are the numbers 1, 2, . . . , k in various permuted orders. The other cases are of course similar. These then are the k! terms xj1 ,1 xj2 ,2 · · · xjk ,k φ(ej1 , ej2 , . . . , ejk ), where j1 , j2 , . . . , jk are the numbers 1, 2, . . . , k in some order. Interchanging the vectors in φ(ej1 , ej2 , . . . , ejk ) repeatedly to make the order ”correct” this term can be written as ±axj1 ,1 xj2 ,2 · · · xjk ,k , where a = φ(e1 , e2 , . . . , ek ) and the sign ± is determined by the parity of the number of uses of antisymmetry (’even’ = +, ’odd’ = −). On the other hand, this is exactly the way the determinant x1,1 x1,2 · · · x1,k x2,1 x2,2 · · · x2,k .. .. .. . . . . . . xk,1 xk,2 · · · xk,k

CHAPTER 4. FORMS

49

is expanded as a sum of products, the only difference is that then a is = 1. The above indicates that in a coordinate representation, certain k-forms of a special type seem to be central. These forms, the so-called elementary k-forms, are defined as follows. Of the indices 1, 2, . . . , n take k indices in increasing order: j1 < j2 < · · · < jk . Then the elementary k-form2 dxj1 ∧ dxj2 ∧ · · · ∧ dxjk is defined by xj ,1 xj ,2 · · · xj ,k 1 1 1 xj ,1 xj ,2 · · · xj ,k 2 2 2 (dxj1 ∧ dxj2 ∧ · · · ∧ dxjk )(r1 , . . . , rk ) = .. .. .. . . . . . . . xjk ,1 xjk ,2 · · · xjk ,k

Thus we get the value of the form by taking elements having the indices j1 , j2 , . . . , jk from r1 , . . . , rk , forming the corresponding k × k determinant, and computing the value of the determinant. As a special case an elementary 0-form is included, it always has the value = 1. Obviously there are n! n = k!(n − k)! k

elementary n-dimensional k-forms. In particular there is one elementary 0-form and one elementary n-form. This makes it possible to ”embed” single scalars in forms in two ways. On the other hand there are n elementary 1-forms, and also n elementary n − 1-forms, which makes it possible to ”embed” vectors in forms via the coefficients, again in two ways (even when n = 2, in a sense!). Note. The definition of the above k-form dxj1 ∧ dxj2 ∧ · · · ∧ dxjk using a determinant actually works even when the indices j1 , j2 , . . . , jk are not in order of magnitude. This possibility is a convenient one to allow. For instance, interchanging dxjl and dxjh interchanges the ith and the hth rows in the determinant and thus changes its sign. Furthermore, if two (or more) of the indices j1 , j2 , . . . , jk are the same, then we naturally agree that the k-form dxj1 ∧ dxj2 ∧ · · · ∧ dxjk has the value 0. Example. n-dimensional elementary 1-forms are simply the component functions. If   x1  x2    r =  ..  ,  .  xn

then (dxj )(r) = xj (j = 1, . . . , n). As determinants these are 1 × 1 determinants.

2

This particular notation is traditional. It does not have much to do with differentials. The symbol ’∧’ is read ”wedge”. The corresponding binary operation, the so-called wedge product (or sometimes the exterior product), is generally available for forms, as will be seen.

CHAPTER 4. FORMS

50

Example. The first component of the cross product r1 × r2 is a 3-dimensional elementary 2-form. If     x1 x2    r1 = y 1 and r2 = y2  , z1 z2 then the first component is

y1 y2 = (dy ∧ dz)(r1 , r2). (r1 × r2 )1 = y1 z2 − z1 y2 = z1 z2

Other components are 2-forms, too:



 (dy ∧ dz)(r1 , r2 ) r1 × r2 =  (dz ∧ dx)(r1 , r2 )  . (dx ∧ dy)(r1 , r2)

Here in the 2-form dz ∧ dx the variables are not in the correct order, cf. the above note. Example. n-dimensional elementary n-forms are simply n × n determinants: (dx1 ∧ dx2 ∧ · · · ∧ dxn )(r1 , r2 , . . . , rn ) = det(r1 , r2 , . . . , rn ). If the variables are not in the correct order and/or are repeated, then (dxj1 ∧ dxj2 ∧ · · · ∧ dxjn )(r1 , r2 , . . . , rn ) = det(rj1 , rj2 , . . . , rjn ), since it does not matter whether it is the rows or the columns that are interchanged. The construct above gives a representation of forms by elementary forms: Theorem 4.2. If φ is an on n-dimensional k-form, then X φ(r1 , . . . , rk ) = aj1 ,j2 ,...,jk (dxj1 ∧ dxj2 ∧ · · · ∧ dxjk )(r1 , . . . , rk ), 1≤j1