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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

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

9. F(x, y) = (y2 cos x + cos y) i + (2y sin xx sin y) j

To determine

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

Explanation

Given data:

The vector field is F(x,y)=(y2cosx+cosy)i+(2ysinxxsiny)j.

Formula used:

For a vector field 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)

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

Calculation:

Compare the vector field F(x,y)=(y2cosx+cosy)i+(2ysinxxsiny)j with F(x,y)=P(x,y)i+Q(x,y)j.

P=y2cosx+cosy (2)

Q=2ysinxxsiny (3)

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

Py=y(y2cosx+cosy)=cosxyy2+y(cosy)=cosx(2y)+(siny) {t(t2)=2t,tcost=sint}=2ycosxsiny

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

Qx=x(2ysinxxsiny)=2yx(sinx)sinyx(x)=2ycosxsiny(1) {t(t)=1,t(sint)=cost}=2ycosxsiny

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

2ycosxsiny=2ycosxsiny

Hence F(x,y)=(y2cosx+cosy)i+(2ysinxxsiny)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)=(y2cosx+cosy)i+(2ysinxxsiny)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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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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