   Chapter 16.3, Problem 2E

Chapter
Section
Textbook Problem

A table of values of a function f with continuous gradient is given. Find  ∫C ∇ f · dr, where C has parametric equationsx = t2 + 1 y = t3 + t 0 ⩽ t ⩽ 1 To determine

To find: The value of Cfdr .

Explanation

Definition:

Refer to FIGURE 1 of section 16.3 in the textbook.

Consider a smooth curve represented by a vector function r(t) , atb .

If f is the differentiable function of two or three variables with gradient vector f is continuous on curve C then Cfdr is,

Cfdr=f[r(b)]f[r(a)] (1)

Express r(t) as x and y coordinates.

r(t)=xi+yj

Substitute t2+1 for x and t3+t for y,

r(t)=(t2+1)i+(t3+t)j (2)

Find the C is smooth curve or not.

Consider r(t) is the vector function of C. If the value of differentiation of r(t) is not equal to 0 then C is a smooth curve.

Differentiate equation (2) with respect to t.

ddt[r(t)]=ddt(t2+1)i+ddt(t3+t)j

r(t)=(2t)i+(3t2+1)j

The value of t limits between 0 and 1.

The value of 3t2+10 for any values lies between 0 and 1. Therefore, C is a smooth curve

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