   Chapter 16.5, Problem 25E

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)25. div(fF) = f div F + F · ∇f

To determine

To prove: The vector field of the form div(fF)=fdivF+Ff .

Explanation

Formula used:

Consider the standard equation of a divergence of vector field.

divF=Px+Qy+Rz (1)

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

Find div(fF) .

div(fF)=div(fP1,Q1,R1)=divfP1,fQ1,fR1

Substitute fP1 for P, fQ1 for Q and fR1 for R in equation (1),

divF=x(fP1)+y(fQ1)+z(fR1)={(fP1x+P1fx)+(fQ1y+Q1fy)+(fR1z+R1f

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