Math 601, Homework 11

9 downloads 4765 Views 55KB Size Report
From Schaum's Outline of Vector Analysis: – Chapter 5 (Surface ... where S is the upper half (z ≥ 0) of the sphere x2 + y2 + z2 = 1. (a) Set up this integral using a ...
Math 601, Homework 11 Due Wednesday, November 28 Solutions should be typed or written neatly and legibly. Answers should be explained. You should reference all your sources, including your collaborators. For more information on writing up homework solutions, see the guidelines at the beginning of Homework 1. Reading assignment: • From Linear Algebra and Vector Calculus at Texas A&M : – Sections 7.1–7.3 • From Schaum’s Outline of Vector Analysis: – Chapter 5 (Surface Integrals) and Chapter 6 (Stokes’ Theorem) Required problems. Turn in a solution for each of the following problems. 1. Consider the curve given by the following parametric equations: x = cos(t)

and

y = sin(2t)

Let D be the region inside this curve and to the right of the y-axis: y

x

ZZ x2 dA.

(a) Use Green’s Theorem to compute D

(b) Use Green’s Theorem to find the area of the region D. 2. Consider the surface integral

ZZ z dA S

where S is the upper half (z ≥ 0) of the sphere x2 + y 2 + z 2 = 1. (a) Set up this integral using a parametrization where t = x and u = y. (b) Set up this integral using a parametrization where t = r and u = θ. (c) Set up this integral using a parametrization where t = θ and u = φ.

3. Use geometric reasoning to evaluate the following surface integrals: (a) The integral

ZZ

¡

x2 + y 2

¢2

dA

S

where S is the surface defined by x2 + y 2 = 4 for 2 ≤ z ≤ 5. (b) The integral

ZZ (x i + y j + z k) · dA S

where S is the sphere of radius 3 centered at the origin. (c) The integral

ZZ (z k) · dA S

where S is the boundary of the region defined by x2 + y 2 ≤ 4 and 1 ≤ z ≤ 3. (d) The integral

ZZ

¡

¢ x2 i + y j + 3z k · dA

S

where S is the boundary of the region defined by 0 ≤ x ≤ 3, 1 ≤ y ≤ 2, and 0 ≤ z ≤ 4. 4. Let S be the surface (r − 5)2 + 9z 2 = 9 (a) Find parametric equations for the surface S. 1 (b) Consider the vector field F(x, y, z) = xz i + yz j − k. Calculate z ZZ the surface integral F · dA. S

Recommended problems. It is recommended that you do many more problems than the required problems. The following list of problems are good practice problems. • From Linear Algebra and Vector Calculus at Texas A&M : – Section 7.1: # 1, 3, 5, 17–23 odd – Section 7.2: # 1–21 odd – Section 7.3: # 1–5 odd, 11 • From Schaum’s Outline of Vector Analysis: – Chapter 5: # 19–24, 58–62, 64–67 – Chapter 6: # 32, 63–66