Math 32B Steven Heilman Please provide complete and well-written solutions to the following exercises. Due March 13, at the beginning of class. Assignment 9 Exercise 1. Let f (x, y) = (1/2) log(x2 + y 2 ) and let C be the circle x2 + y 2 = a2 . Let en denote the unit exterior normal of C in the plane. • Evaluate the flux integral I (∇f ) · en ds. C • Let K be a continuously differentiable simple closed curve in the plane that does not intersect the origin (0, 0). Suppose we give K the counterclockwise orientation. Using the Divergence Theorem, show that I (∇f ) · en ds K has two possible values, depending on whether (0, 0) lies inside K or outside K. (Hint: It may be helpful to consider C which is a very large circle containing K.) Exercise 2. Consider the vector field F (x, y, z) = (−y/(x2 + y 2 ), x/(x2 + y 2 ), z). Show that H curl(F ) = 0, but C F · T ds is nonzero where C is the circle x2 + y 2 = 1, z = 0. Explain why this does not contradict Stokes’ Theorem. Exercise 3. Let F : R3 → R3 be a vector field. Verify that div(curl(F )) = 0. Can you use this identity to conclude anything about the flux of the field curl(F ) across a surface S, where S is the boundary of a region D in Euclidean space R3 ? Justify your answer. Exercise 4 (Properties of divergence). Let g : R3 → R be a function, and let F : R3 → R3 be a vector field. Verify: div(gF ) = (g)div(F ) + (∇g) · (F ) Exercise 5. Let F (x, y, z) = (y − x, z − y, y − x) be a vector field. Let D denote the cube bounded by the planes x = 1, x = −1, y = 1, y = −1, and z = 1, z = −1. Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. Exercise 6. Let F (x, y, z) = (6x2 + 2xy, 2y + x2 z, 4x2 y 3 ). Let D be the region cut from the first octant by the cylinder x2 + y 2 = 4 and by the plane z = 3. Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. 2 Exercise 7. Let F (x, y, z) = (x, y, z) be a vector field, let D be a region in Euclidean space R3 with boundary S. Let en denote the unit exterior normal to S. Show that the volume of D is equal to ZZ 1 F · en dS 3 S Exercise 8 (Green’s formulas). Let f, g : R3 → R be functions. Let D be a region in R3 with boundary S. Let en denote the unit exterior normal to S. Verify the following equalities ZZ ZZZ • f ∇g · en dS = (f div(∇g) + ∇f · ∇g) dV D Z ZS ZZZ • (f ∇g − g ∇f ) · en dS = (f div(∇g) − g div(∇f )) dV S D Exercise 9. Suppose f : R3 → R is twice continuously differentiable, and ∂ 2f ∂ 2f ∂ 2f + + = 0. ∂x2 ∂y 2 ∂z 2 Let S(r) denote the sphere of radius r centered at the origin. Define ZZ 1 g(r) = f dS. 4πr S(r) Prove that (d/dr)g(r) = 0. (Hint: first change variables so that g is an integral over S(1). Then, differentiate under the integral and use the divergence theorem.)