Chapter 16, Problem 16RCC

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

In what ways are the Fundamental Theorem for Line Integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem similar?

To determine

**To describe**: The similarities among the Fundamental theorem for line integrals, Green’s theorem, Stokes’ theorem, and the Divergence theorem.

Explanation

Consider a smooth curve *C* which is given by vector
*f* is differentiable and the gradient of *f*
*C*. According to the Fundamental theorem of line integrals, the line integral over the curve *C* is,

Here,

*a* and *b* are the limits of domain.

Consider a positively oriented curve *C* which is piece-wise smooth, simple closed curve in plane with domain *D*. According to the Greens theorem,

Here,

*P*,

*Q*, and

*P* and *Q* have continuous partial derivatives

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