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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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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Chapter 16 Solutions

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Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 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