   Chapter 16.3, Problem 28E

Chapter
Section
Textbook Problem

Let F = ∇ f, where f(x, y) = sin(x − 2y). Find curves C1 and C2 that are not closed and satisfy the equation.(a) ∫ C 1 F · dr = 0 (b) ∫ C 1 F · dr = 1

(a)

To determine

To find: The curve C1 such that C1Fdr=0 .

Explanation

Given data:

Vector function f(x,y)=sin(x2y) .

Formula used:

Consider vector function r(t) , atb with a smooth curve C. Consider f is a differentiable function two or three variables of gradient function f and is continuous on curve C. Then,

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

Here,

a is starting point, and

b is ending point.

Write the expression for vector function r(t) of curve C from a(x1,y1) to b(x2,y2) .

r(t)=(x2x1)ti+(y2y1)tj (2)

Here,

t is parameter.

Write the expression for vector function.

f(x,y)=sin(x2y) (3)

Write the expression for f .

f=fx(x,y)i+fy(x,y)j (4)

Apply partial differentiation with respect to x on both sides of equation (3).

fx(x,y)=x(sin(x2y))=cos(x2y)(10){t(sint)=cost,t(t)=1}=cos(x2y)

Apply partial differentiation with respect to y on both sides of equation (3)

(b)

To determine

To find: The curve C1 such that C1Fdr=1 .

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