   Chapter 16.5, Problem 26E

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)26. curl(fF) = f curl F + (∇f) × F

To determine

To prove: The vector field of the form curl(fF)=fcurlF+(f)×F .

Explanation

Formula used:

Consider the standard equation of a curl F.

curlF=|ijkxyzPQR| (1)

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

Find curl(fF) .

curl(fF)=curl(fP1,Q1,R1)=curlfP1,fQ1,fR1

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

curlF=|ijkxyzfP1fQ1fR1|={[y(fR1)z(fQ1)]i[z(fP1)x(fR1)]j+[x(fQ1)y(fP1)]k}={[fy(R1)+R1fyfy(Q1)Q1fy]i[fz(P1)+P1fzfx(R1)R1fx]j+[fx(Q1)+Q1fxfy(P1)P1fy]k} {t(uv)=ut(v)+vt(u)}={[fy(R1)fy(Q1)]i[fz(P1)fx

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

Draw a polygon for the distribution of scores shown in the following table. X f 6 2 5 5 4 3 3 2 2 1

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

7-12 Find the limit. limxsin4xx

Calculus (MindTap Course List)

Prove each identity. tanx2=tanxsecx+1

Trigonometry (MindTap Course List)

Which is the best graph of r = 1 − sin θ for 0 ≤ θ ≤ π?

Study Guide for Stewart's Multivariable Calculus, 8th 