   Chapter 16, Problem 5P

Chapter
Section
Textbook Problem

Prove the following identity: ∇(F · G) = (F · ∇)G + (G · ∇)F + F × curl G + G × curl F

To determine

To prove: The identity (FG)=(F)G+(G)F+F×curl G+G×curl F .

Explanation

Formula used:

Write the expression for .

=xi+yj+zk (1)

Write the required differential and integration formulae to evaluate the given integral.

ddxxn=nxn1[f(x)]ndx=[f(x)]n+1n+1

Write the expression for curlF .

curlF=|ijkxyzPQR| (2)

Consider the expression for F as follows.

F=P1i+Q1j+R1k (3)

Consider the expression for G as follows.

G=P2i+Q2j+R2k (4)

Find the value of F .

F=(P1i+Q1j+R1k)(xi+yj+zk)=P1x+Q1y+R1z

Find the value of (F)G .

(F)G=(P1x+Q1y+R1z)(P2i+Q2j+R2k)={(P1P2x+Q1P2y+R1P2z)i+(P1Q2x+Q1Q2y+R1Q2z)j+(P1R2x+Q1R2y+R1R2z)k}=(FP2)i+(FQ2)j+(FR2)k

Find the value of G .

G=(P2i+Q2j+R2k)(xi+yj+zk)=P2x+Q2y+R2z

Find the value of (G)F .

(G)F=(P2x+Q2y+R2z)(P1i+Q1j+R1k)={(P2P1x+Q2P1y+R2P1z)i+(P2Q1x+Q2Q1y+R2Q1z)j+(P2R1x+Q2R1y+R2R1z)k}=(GP1)i+(GQ1)j+(GR1)k

Write the expression for F(x,y,z) .

F(x,y,z)=Pi+Qj+Rk (5)

Compare equation (3) and (5).

P=P1Q=Q1R=R1

Substitute P1 for P , Q1 for Q and R1 for R in equation (2),

curlF=|ijkxyzP1Q1R1|={i[R1yQ1z]j[R1xP1z]+k[Q1xP1y]}

Find the value of G×curlF

G×curlF={P2i+Q2j+R2k}×{i[R1yQ1z]j[R1xP1z]+k[Q1xP1y]}=[ijkP2Q2R2(R1yQ1z)(R1xP1z)(Q1xP1y)]={[Q2(Q1xP1y)+R2(R1xP1z)]i[P2(Q1xP1y)R2(R1yQ1z)]j+[(P2)(R1xP1z)Q2(R1yQ1z)]k}={[Q2Q1xQ2P1y+R2R1xR2P1z]i+[P2P1yP2Q1x+R2R1yR2Q1z]j+[P2P1zP2R1xQ2R1y+Q2Q1z]k}

Write the expression for G(x,y,z) .

G(x,y,z)=Pi+Qj+Rk (6)

Compare equation (4) and (6).

P=P2Q=Q2R=R2

Modify equation (2) as follows

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