Math 6B

Worksheet 6

Winter 2025

Due Wednesday, Feb 12, at 5pm.

1. Let f, g : R n 7→ R and F, G : R n 7→ R n . Verify the following identities.

Assume that these functions are differentiable with continuous derivatives to the order that is shown in the given identity.

Notation. ∇ · ∇f = ∇2 f = ∆f.

(a) ∇(fg) = f∇g + g∇f

(b) ∇ · (F + G) = ∇ · F + ∇ · G

(c) ∇ · (fF) = f∇ · F + F · ∇f

(d) ∇ × (f∇g) = ∇f × ∇g

(e) Curl of gradient is 0: ∇ × ∇f = 0 (Here 0, the bold face zero, is the zero vector)

(f) Divergence of curl is 0: ∇ · (∇ × F) = 0

(g) First Green’s identity:

Second Green’s identity:

2. (Stewart) Let S be an oriented smooth surface bounded by a simple, closed, smooth bound-ary curve C with positive orientation, and f, g : R 3 7→ R be twice-differentiable functions with continuous partial derivatives. Show that

3. Show that the Green’s theorem is a special case of the Stokes’ theorem.

4. (Stewart) Let F(x1, x2, x3) = −x1x2 i + x3 j + 2x2 2 k. Let A = (−2, 0, 0), B = (0, 0, 0), C = (0, 3, 2), and D = (−2, 3, 2). An object moves on line segments from A to B to C to D, and then back to A. Evaluate the work done by the object

(a) directly,

(b) using Stokes’ theorem.

5. (Stewart) Let F(x1, x2, x3) = ⟨F1(x1, x2, x3), F2(x1, x2, x3), F3(x1, x2, x3)⟩ be a vector field where every Fi , for 1 ≤ i ≤ 3, has continuous partial derivatives. Let S be a sphere in the domain of F. Show that