   Chapter 16, Problem 5RCC

Chapter
Section
Textbook Problem

State the Fundamental Theorem for Line Integrals.

To determine

To state: The fundamental theorem of line integrals.

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))

Here,

a and b are the limits of domain.

The Fundamental theorem of line integrals is applicable for a conservative vector field.

In case the function f has two variables, consider the start point of curve C as A(x1,y1) and the terminal point as B(x2,y2) in two-dimensional plane, then the Fundamental theorem of line integrals is,

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