Chapter 16, Problem 4RCC

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) Define the line integral of a vector field **F** along a smooth curve *C* given by a vector function **r**(*t*).

(b) If **F** is a force field, what does this line integral represent?

(c) If **F** = ⟨*P*, *Q, R*⟩, what is the connection between the line integral of **F** and the line integrals of the component functions *P*, *Q*, and *R*?

**(a)**

To determine

**To define**: The line integral of vector field
*C*.

Explanation

The integrals which are done over a curve instead of a interval are referred as line integrals.

Consider a smooth curve *C*,

The first derivative of vector function
*C* is a smooth curve. Divide the parametric interval of curve *C* which is
*n* subintervals as

Hence, the parametric equations at simple point
*i*th subarc are,

Consider a vector field
*C*. Then the line integral of function is equal to the sum of product of value of *f* at point

The line integral of vector field
*C* is,

Consider the length of subarc
*L*.

Re-modify the equation (1).

Write the expression for length of *C* on

Substitute
*L* in equation (2),

Write the expression for

Here,

Write the expression for unit tangent vector

Here,

**(b)**

To determine

**To explain**: If the vector field

**(c)**

To determine

**To explain**: The relation between the line integral of
*P*, *Q*, and *R*.

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