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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Show that F is a conservative and use this fact to evaluate ∫C F · dr along the given curve.

13. F(x, y) = (4x3y2 – 2xy3) i + (2x4y – 3x2y2 + 4y3) j, C: r(t) = (t + sin πt) i + (2t + cos πt) j, 0 ⩽ t ⩽ 1

To determine

To show: F is a conservative vector field and evaluate the value of CFdr.

Explanation

Given data:

F(x,y)=(4x3y22xy3)i+(2x4y3x2y2+4y3)j (1)

C:r(t)=(t+sinπt)i+(2t+cosπt)j,0t1 (2)

Formula used:

Consider the standard equation of curl F for F=Pi+Qj+Rk

curlF=|ijkxyzPQR| (3)

If the curl of a vector field F is zero, then F is said to be a conservative vector field.

The relation between the potential function f and vector field F is F=f, where f=fx(x,y)i+fy(x,y)j.

Obtain the value of curlF.

Substitute (4x3y22xy3) for P, (2x4y3x2y2+4y3) for Q and 0 for R in equation (3).

curlF=|ijkxyz(4x3y22xy3)(2x4y3x2y2+4y3)0|={[y(0)z(2x4y3x2y2+4y3)]i[x(0)z(4x3y22xy3)]j+[x(2x4y3x2y2+4y3)y(4x3y22xy3)]k}=(00)i(00)j+(8x3y6xy2+08x3y+6xy2)k=(00)i(00)j+(0)k

On simplification,

curlF=0

Since curlF=0, F is a conservative vector and the domain of F is 2.

Thus, F is a conservative vector field.

Substitute fx(x,y)i+fy(x,y)j for f in F=f.

F=fx(x,y)i+fy(x,y)j (4)

Compare the equation (4) and equation (1).

fx(x,y)=4x3y22xy3 (5)

fy(x,y)=2x4y3x2y2+4y3 (6)

Integrate equation (5) with respect to x.

f(x,y)=(4x3y22xy3)dx=4x44y22x22y3+g(y)

f(x,y)=x4y2x2y3+g(y) (7)

Apply partial differentiation with respect to y on both sides of equation (7)

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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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