   Chapter 16.2, Problem 49E

Chapter
Section
Textbook Problem

If C is a smooth curve given by a vector function r(t), a ⩽ t ⩽ b, and v is a constant vector, show that∫C v · dr = v · [r(b) − r(a)]

To determine

To show: The expression Cvdr=v[r(b)r(a)] .

Explanation

Given data:

The curve C consists of a position vector function r(t) .

The limits of scalar parameter is given as atb .

v is considered as constant vector.

Formula used:

Write the expression to find Cvdr as follows.

Cvdr=abvr(t)dt (1)

Write the expression to find r(t) of the object.

r(t)=ddt[r(t)] (2)

Consider the position vector r(t) as follows.

r(t)=x(t),y(t),z(t)

Consider the constant vector function v as follows.

v=v1,v2,v3

Calculation of r(t) :

Substitute x(t),y(t),z(t) for r(t) in equation (2),

r(t)=ddt[x(t),y(t),z(t)]=ddt[x(t)],ddt[y(t)],ddt[z(t)]=x(t),y(t),z(t)

Calculation of Cvdr :

Substitute v1,v2,v3 for v , x(t),y(t),z(t) for r(t) , a for a , and b for b in equation (1),

Cvdr=abv1,v2,v3x(t),y(t),z(t)dt=ab[v1x(t)+v2y(<

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