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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
True or False? In Exercises 43-46, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If F is conservative in a region R bounded by a simple closed path and C lies within R, then is independent of path.