Cartesian, Cylindrical Polar, and Spherical Polar Coordinates

Cartesian, Cylindrical Polar, and Spherical Polar Coordinates

z

r r

y

x

φ

Spherical Polar

x

ρ cos φ

eˆφ = − sin φ iˆ + cos φ jˆ

r sin θ cos φ r sin θ sin φ r cos θ eˆr = sin θ cos φ iˆ + sin θ sin φ jˆ + cos θ kˆ eˆθ = cos θ cos φ iˆ + cos θ sin φ jˆ − sin θ kˆ

eˆz = kˆ

eˆφ = − sin φ iˆ + cos φ ˆj

z=

z

Unit Vectors (orthonormal)

eˆ x = iˆ eˆ = jˆ y

eˆz = kˆ

r dr = dx iˆ + dy ˆj + dz kˆ

ρ sin φ z eˆρ = cos φiˆ + sin φ jˆ

r dr = dρ eˆρ + ρ dφ eˆφ + dz eˆz

(ds )2 = (dx )2 + (dy )2 + (dz )2 (ds )2 = (dρ)2 + ρ2 (dφ)2 + (dz )2

Volume Element

PH461/PH561, Fall 2008

φ

Cylindrical Polar

y

r2 2 dr ≡ (ds )

ρ

r

Cartesian x= y=

Infinitesimal Displacement

θ

z

dV = dx dy dz

dV = ρ dρ dφ dz

page 1 of 2

r dr = dr eˆr + r dθ eˆθ + r sin θ dφ eˆφ

(ds )2 = (dr )2 + r 2 (dθ)2 + r 2 sin2 θ (dφ)2 dV = r 2 sin θ dr dθ dφ

©2008 William W. Warren, Jr.

Vector Operators in Cartesian, Cylindrical Polar, and Spherical Polar Coordinates r In the table below, Φ is a scalar function of the spatial coordinates (not to be confused with the azimuthal angle φ) and a is a vector field, i.e. a vector whose components depend on the spatial coordinates. The vectors iˆ, jˆ, eˆρ, eˆθ etc. are unit

vectors pointing in the direction of increasing values of the respective coordinates.

Operator r ∇Φ

r r ∇⋅a

r r ∇×a

∇ 2Φ

Cartesian

Cylindrical Polar

Spherical Polar

∂Φ ˆ ∂Φ ˆ ∂Φ ˆ i + j+ k ∂x ∂y ∂z

∂Φ 1 ∂Φ ∂Φ eˆρ + eˆφ + eˆz ∂ρ ρ ∂φ ∂z

∂Φ 1 ∂Φ 1 ∂Φ eˆr + eˆθ + eˆφ ∂r r ∂θ r sin θ ∂φ

∂a x ∂a y ∂a z + + ∂x ∂y ∂z

1 ∂ 1 ∂a φ ∂a z ρa ρ + + ρ ∂ρ ρ ∂φ ∂z

iˆ ∂ ∂x ax

ˆj ∂ ∂y ay

kˆ ∂ ∂z az

∂ 2Φ ∂ 2Φ ∂ 2Φ + + ∂z 2 ∂y 2 ∂x 2

PH461/PH561, Fall 2008

(

)

eˆρ ρeˆφ ∂ 1 ∂ ρ ∂ρ ∂φ aρ ρaφ

eˆz ∂ ∂z az

1 ∂ ⎛ ∂Φ ⎞ 1 ∂ 2 Φ ∂ 2 Φ ⎟+ ⎜ρ + ρ ∂ρ ⎜⎝ ∂ρ ⎟⎠ ρ2 ∂φ2 ∂z 2

page 2 of 2

( )

∂aφ 1 ∂ 2 1 ∂ (sin θaθ ) + 1 r ar + 2 r sin θ ∂θ r sin θ ∂φ r ∂r

eˆr 1 ∂ 2 r sin θ ∂r ar

reˆθ ∂ ∂θ raθ

r sin θeˆφ ∂ ∂φ r sin θaφ

1 1 ∂ ⎛ 2 ∂Φ ⎞ 1 ∂ 2Φ ∂ ⎛ ∂Φ ⎞ sin θ + + r ⎟ ⎜ ⎟ ⎜ ∂θ ⎠ r 2 sin2 θ ∂φ2 r 2 ∂r ⎝ ∂r ⎠ r 2 sin θ ∂θ ⎝

©2008 William W. Warren, Jr.