Proof Let , where M and N have continuous first partial derivatives in a simply connected region R. Prove that if C is simple, smooth, and closed, and then .
For unlimited access to Homework Help, a Homework+ subscription is required.
Proof In Exercises 51 and 52, prove the identity, where R is a simply connected region with piecewise smooth boundary C. Assume that the required partial derivatives of the scalar functions f and g are continuous. The expressions and are the derivatives in the direction of the outward normal vector N of C and are defined by and .
Green’s second identity:
(Hint: Use Green’s first identity from Exercise 51 twice.)
Proof In Exercises 51 and 52, prove the identity, where R is a simply connected region with piecewise smooth boundary C. Assume that the required partial derivatives of the scalar functions f and g are continuous. The expressions
g are the derivatives in the direction of the outward normal vector N of C and are defined by
Green’s first identity:
[Hint: Use the second alternative form of Green’s Theorem and the property div
HOW DO YOU SEE IT? The figure shows a region R bounded by a piecewise smooth simple closed path C.
(a) Is R simply connected? Explain.
(b) Explain why , where f and g are differentiable functions.