   Chapter 16.5, Problem 28E

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)28. div(∇f × ∇g) = 0

To determine

To prove: The vector field of the form div(f×g)=0 .

Explanation

Formula used:

Consider the standard equation of a divergence of vector field.

divF=F (1)

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

Modify equation (1) to find div(f×g) .

div(f×g)=(f×g)=|xyzfxfyfzgxgygz|=x|fyfzgygz|y|fx

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