   Chapter 16.5, Problem 29E

Chapter
Section
Textbook Problem

Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then fF, F · G, and F × G are defined by(fF)(x, y, z) = f(x, y, z) F(x, y, z)(F · G)(x, y, z) = F(x, y, z) · G(x, y, z)(F × G)(x, y, z) = F(x, y, z) × G(x, y, z)29. curl(curl F) = grad(div F) − ∇2F

To determine

To prove: The vector field of the form curl(curlF)=grad(divF)2F .

Explanation

Formula used:

Consider the standard equation of an curl F.

curlF=|ijkxyzPQR|

Consider the standard equation of a divergence of vector field.

divF=Px+Qy+Rz

Consider F(x,y,z)=P1i+Q1j+R1k and G(x,y,z)=P2i+Q2j+R2k .

Find curlF .

curlF=|ijkxyzP1Q1R1|=[R1yQ1z]i+[P1zR1x]j+[Q1xP1y]k

Find curl(curlF) .

curl(curlF)=|ijkxyzR1yQ1zP1zR1xQ1xP1y|={[y(Q1xP1y)z(P1zR1x)]i+[z(R1yQ1z)x(Q1xP1y)]j+[x(P1zR1x)y(R1yQ1z)]k}={[(2Q1xy2P1y2)(2P1z22R1xz)]i+[(2R1yz2Q1z2)(2Q1x22P1xy)]j+[(2P1xz2R1x2)(2R1y2Q1yz)]k}

curl(curlF)={[(2Q1xy2P1y2)2P1z2+2R1xz]i+[(2R1yz2Q1z2)2Q1x2+2P1xy]j+[(2P1xz2R1x2)2R1y2+Q1yz]k} (1)

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