   Chapter 16.4, Problem 29E

Chapter
Section
Textbook Problem

If F is the vector field of Example 5, show that ∫C F · dr = 0 for every simple closed path that does not pass through or enclose the origin.

To determine

To show: The value of CFdr for F(x,y)=(yi+xj)x2+y2 is 0.

Explanation

Given data:

Field vector is F(x,y)=(yi+xj)x2+y2 .

Formula used:

Green’s Theorem:

Consider a positively oriented curve C which is piece-wise smooth, simple closed curve in plane with region D. Then the line integration of vector field F(x,y)=P(x,y),Q(x,y) over curve C is,

CFdr=D(QxPy)dA (1)

Here,

A is area,

Py is continuous first-order partial derivative of P,

Qx is continuous first-order partial derivative of Q, and

P and Q have continuous partial derivatives.

Consider the field as F(x,y)=yi+xjx2+y2 .

The curve C is simple and it is not passing through the origin or enclosing origin. Hence, the region of curve C does not contain origin, but Green’s Theorem is applicable.

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

Simplify the equation.

Py=y2x2(x2+y2)2

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

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