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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

If F and G are vector fields whose component functions have continuous first partial derivatives, show that curl(F × G) = F div GG div F + (G · ∇)F − (F · ∇)G

To determine

To Show: If F and G are vector fields whose component functions have continuous first partial derivatives then curl(F×G)=F div GG div F+(G)F(F)G .

Explanation

Formula used:

Write the expression for curlF .

curlF=|ijkxyzPQR| (1)

Write the expression for divF .

divF=Pxi+Qyj+Rzk (2)

Write the expression for F×G .

F×G=|ijkP1Q1R1P2Q2R2| (3)

Consider two vector fields as follows.

F=P1i+Q1j+R1kG=P2i+Q2j+R2k

Modify equation (2) as follows.

divF=P1xi+Q1yj+R1zk

Modify equation (2) as follows.

divG=P2xi+Q2yj+R2zk

Find the value of F×G using equation (3).

F×G=|ijkP1Q1R1P2Q2R2|=(Q1R2Q2R1)i(P1R2P2R1)j+(P1Q2P2Q1)k

Find the value of curl(F×G) using equation (1).

curl(F×G)=|ijkxyz(Q1R2Q2R1)(P1R2P2R1)(P1Q2P2Q1)|=|ijkxyz(Q1R2Q2R1)(P2R1P1R2)(P1Q2P2Q1)|

curl(F×G)={[y(P1Q2P2Q1)z(P2R1P1R2)]i[x(P1Q2P2Q1)z(Q1R2Q2R1)]j+[x(P2R1P1R2)y(Q1R2Q2R1)]k} (4)

Find the value of (G)F(F)G .

(G)F(F)G={[(P2P1x+Q2P1y+R2P1z)i+(P2Q1x+Q2Q1y+R2Q1z)j+(P2R1x+Q2R1y+R2R1z)k][(P1P2x+Q1P2y+R1P2z)i+(P1Q2x+Q1Q2y+R1Q2z)j+(P1R2x+Q1R2y+R1R2z)k]} (5)

Find the value of Fdiv GGdiv F .

Fdiv GGdiv F={(P1i+Q1j+R1k)(P2xi+Q2yj+R2zk)(P2i+Q2j+R2k)(P1xi+Q1yj+R1zk)}

Fdiv GGdiv F={[P1(P2x+Q2y+R2z)i+Q1(P2x+Q2y+R2z)j+R1(P2x+Q2y+R2z)k][P2(P1x+Q1y+R1z)i+Q2(P1x+Q1y+R1z)j+R2(P1x+Q1y+R1z)k]} (6)

Find the value of F div GG div F+(G)F(F)G using equation (5) and (6)

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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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