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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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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Chapter 16 Solutions

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