From Linear Algebra and Vector Calculus at Texas A&M: – Sections 3.2–3.4, 6.1–
6.3. • From Schaum's Outline of Vector Analysis: – Chapter 4, Chapter 5 (Line ...
Math 601, Homework 10 Due Friday, November 16 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 3.2–3.4, 6.1–6.3 • From Schaum’s Outline of Vector Analysis: – Chapter 4, Chapter 5 (Line Integrals), Chapter 6 (Green’s Theorem in the Plane) Required problems. Turn in a solution for each of the following problems. 1. Consider the vector field F(x, y) = (2x + 2y)i + (2x − y)j. (a) Find the parametric equations for the flow line of F beginning at the point (0, 5). (b) Find div(F). (c) Find rot(F). 2. Find a vector field F whose flow lines are the parametric curves x=t
y = C(t2 + 1)
x2 y 2 + = 1 in the first quad3. (a) Let C be the portion of the ellipse 4 9 rant. Evaluate the following integral: Z xy ds C
(b) Let C be the curve in R3 described by the equations ρ = θ π and φ = . with endpoints (x, y, z) = (0, 0, 0) and (x, y, z) = 4√ √ (π 2, 0, π 2). Evaluate the following integral: Z x y dx + dy + z 3 dz z C z
4. Use geometric reasoning to evaluate the following line integrals: (a) The integral
Z
¡
x2 + y 2
¢2
ds
C
where C is the circle of radius 3 centered at the origin. (b) The integral
Z −ry dx + rx dy C
where C is the circle of radius 2 centered at the origin. (c) The integral
Z x dx + y dy + z dz C
where C is any curve on the sphere x2 + y 2 + z 2 = 9. 5. Consider the vector field F = (2x + y) i + (x + 3y 2 ) j. (a) Show that F is conservative. (b) Find a function f such that ∇f = F. Z (c) Evaluate the integral F · ds where C is the curve y = sin(x2 ) C √ from (0, 0) to ( π, 0). 6. Let C be the circle of radius 1 centered at the origin and oriented counterclockwise. Use Green’s Theorem to evaluate the following integral: I (ey − y 3 ) dx + (xey + x3 ) dy C
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 3.1: # 1–5 odd – Section 3.2: # 1–7 odd, 11 – Section 3.3: # 1–9 odd, 17–23 odd – Section 3.4: # 1–13 odd – Section 6.1: # 1–21 odd – Section 6.2: # 1–13 odd, 17, 19 – Section 6.3: # 1–17 odd • From Schaum’s Outline of Vector Analysis: – Chapter 1: # 30 – Chapter 4: # 1, 3, 6, 7, 10, 11, 12, 15, 16, 22–24, 32, 58 – Chapter 5: # 6–9, 12, 14b, 37–41, 43–45, 47, 49, 51, 52 – Chapter 6: # 6–9, 13, 37–44, 46, 47, 49, 50
