   Chapter 16.2, Problem 50E

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 r · dr = 1 2 [|r(b)|2 − |r(a)|2]

To determine

To show: The expression Crdr=12[|r(b)|2|r(a)|2] .

Explanation

Given data:

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

The limits of scalar parameter is given as atb .

Formula used:

Write the expression to find Crdr as follows.

Crdr=abr(t)r(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)

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 Crdr :

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

Crdr=abx(t),y(t),z(t)x(t),y(t),z(t)dt=ab[x(t)x(t)+y(t)y(t)+z(t)z(t)]dt=ab

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