   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

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Simplify the expressions in Exercises 97106. (xy)1/3(yx)1/3

Finite Mathematics and Applied Calculus (MindTap Course List)

109-110 Find f(x). f(x)=1xessds

Calculus (MindTap Course List)

Find dy/dx by implicit differentiation. 9. x2x+y=y2+1

Single Variable Calculus: Early Transcendentals

f(x) = 12x x3 has a local maximum at: a) x = 0 b) x = 2 c) x = 2 d) f does not have a local maximum

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

In Exercises 21-26, use the laws of logic to prove the propositions. p(qr)(pq)(pr)

Finite Mathematics for the Managerial, Life, and Social Sciences 