   Chapter 16.5, Problem 27E

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)27. div(F × G) = G · curl F − F · curl G

To determine

To prove: The vector field of the form div(F×G)=GcurlFFcurlG .

Explanation

Formula used:

Consider the standard equation of a divergence of vector field.

divF=F (1)

Consider the standard equation of a curl F.

curlF=|ijkxyzPQR|

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)=|xyzP1Q1R1P2Q2R2|=x|Q1R1Q2R2|y|P1R1P2R2|+z|P1Q1P2Q2|=x(Q1R2Q2R1)y(P1R2P2R1)+z(P1Q2P2Q1)

Simplify the equation.

div(F×G)={[x(Q1R2)x(Q2R1)][y(P1R2)y(P2R1)]+[z(P1Q2)z(P2Q1)]}={[Q1x(R2)+R2x(Q1)Q2x(R1)R1x(Q2)][P1y(R2)+R2y(P1)P2y(R1)R1y(P2)]+[P

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Solve for y in terms of x: 4x5y=10

Elementary Technical Mathematics

Evaluate the integral. 03(1+6w210w4)dw

Single Variable Calculus: Early Transcendentals

Given: BEDACDDA Prove: 12

Elementary Geometry for College Students 