   Chapter 16, Problem 20RE

Chapter
Section
Textbook Problem

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