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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

Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇ f.

8. F(x, y) = (2xy + y−2) i + (x2 − 2xy−3) j, y > 0

To determine

Whether F is a conservative vector field and find corresponding function f such that F=f .

Explanation

Given data:

Vector field is F(x,y)=(2xy+y2)i+(x22xy3)j .

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)=(2xy+y2)i+(x22xy3)j with F(x,y)=P(x,y)i+Q(x,y)j .

P=2xy+y2 (2)

Q=x22xy3 (3)

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

Py=y(2xy+y2)=2xyy+y(y2)=2x(1)+(2y3) {t(t)=1,ttn=ntn1}=2x2y3

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

Qx=x(x22xy3)=x(x2)2y3x(x)=2x2y3(1) {t(t)=1,t(t2)=2t}=2x2y3

Substitute 2x2y3 for Py and 2x2y3 for Qx in equation (1),

2x2y3=2x2y3

Hence F(x,y)=(2xy+y2)i+(x22xy3)j is conservative vector field.

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

Write the relation between the potential function f and vector field F .

f=F

Substitute fx(x,y)i+fy(x,y)j for f ,

F=fx(x,y)i+fy(x,y)j

Compare the equation F=fx(x,y)i+fy(x,y)j with F(x,y)=(2xy+y2)i+(x22xy3)j

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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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