   Chapter 16.3, Problem 35E

Chapter
Section
Textbook Problem

Let F(x, y) = − y i + x j x 2 + y 2 (a) Show that ∂ P / ∂ y   =   ∂ Q / ∂ x .(b) Show that ∫C F · dr is not independent of path. [Hint: Compute ∫ C 1 F · dr and ∫ C 2 F · dr, where C1 and C2 are the upper and lower halves of the circle x2 + y2 = 1 from (1, 0) to (‒1, 0).] Does this contradict Theorem 6?

(a)

To determine

To show: Py=Qx for vector filed F(x,y)=yi+xjx2+y2 .

Explanation

Given data:

Vector field F(x,y)=yi+xjx2+y2 .

Formula used:

Consider a vector field as F(x,y)=P(x,y)i+Q(x,y)j . The condition for vector field F being a conservative field is,

Py=Qx (1)

Here,

Py is continuous first-order partial derivative of P, and

Qx is continuous first-order partial derivative of Q,

Compare the vector field F(x,y)=P(x,y)i+Q(x,y)j with F(x,y)=yi+xjx2+y2 .

P=yx2+y2 (2)

Q=xx2+y2 (3)

Apply partial differentiation with respect to y on both sides of equation (2).

Py=y(yx2+y2)=(y(y))(x2+y2)(y)(y(x2+y2))(x2+y2)2 {t(uv)=uvuvv2}=(1)(x2+y2)+y(0+2y)(x2+y2)2 {t(k)=0,t(t)=1,t(t2)=2y}=x2y2+2y2(x2+y2)2

Py=y2x2(x2+y2)2

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

(b)

To determine

To show: The line integral Cfdr is not independent of path.

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