Chapter 16.8, Problem 20E

Suppose S and C satisfy the hypotheses of Stokes’ Theorem and f, g have continuous second-order partial derivatives. Use Exercise 24 and 26 in Section 16.5 to show the following.

(a) ${\displaystyle {\int }_C\left(f\nabla g\right)}\cdot dr={\displaystyle {\iint }_S\left(\Delta f\times \nabla g\right)}\cdot dS$

(b) ${\displaystyle {\int }_C\left(f\nabla f\right)}\cdot dr=0$

(c) ${\displaystyle {\int }_C\left(f\nabla g+g\nabla f\right)}\cdot dr=0$