   Chapter 16, Problem 16RCC

Chapter
Section
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 r(t) in a region D={atb . Consider function f is differentiable and the gradient of f f is continuous on curve C. According to the Fundamental theorem of line integrals, the line integral over the curve C is,

Cfdr=f(r(b))f(r(a)) (1)

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,

CPdx+Qdy=D(QxPy)dA (2)

Here,

Py is continuous first-order partial derivative of P,

Qx is continuous first-order partial derivative of Q, and

P and Q have continuous partial derivatives

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