   Chapter 16.5, Problem 23E

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)23. div(F + G) = div F + div G

To determine

To prove: The vector field of the form div(F+G)=divF+divG .

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(F+G) .

div(F+G)=div[(P1i+Q1j+R1k)+(P2i+Q2j+R2k)]=div(P1,Q1,R1+P2,Q2,R2)=divP1+P2,Q1+Q2,R1+R2

Substitute P1+P2 for P , Q1+Q2 for Q and R1+R2 for R in equation (1),

div(F+G)=x(P1+P2

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