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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
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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Chapter 16 Solutions

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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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