   Chapter 16.5, Problem 39E

Chapter
Section
Textbook Problem

We have seen that all vector fields of the form F = ∇g satisfy the equation curt F = 0 and that all vector fields of the form F = curl G satisfy the equation div F = 0 (assuming continuity of the appropriate partial derivatives). This suggests the question: are there any equations that all functions of the form f = div G must satisfy? Shoes that the answer to this question is “No” by proving that every continuous function f on ℝ3 is the divergence of sonic vector field.[Hint: Let G(x, y, z) = ⟨g(x, y, z), 0, 0⟩, where g ( x ,   y ,   z )  = ∫ 0 x f ( t , y , z )   d t .]

To determine

To show: The every continuous function f on 3 is the divergence of some vector field.

Explanation

Given data:

G(x,y,z)=g(x,y,z),0,0

g(x,y,z)=0xf(t,y,z)dt

Formula used:

Consider the standard equation of a divergence of vector field for F=Pi+Qj+Rk

divF=Px+Qy+Rz (1)

Modify equation (1).

divG=Px+Qy+Rz

Substitute g(x,y,z) for P , 0 for Q and 0 for R in equation (1),

divG=

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

Find more solutions based on key concepts 