   Chapter 16.5, Problem 32E

Chapter
Section
Textbook Problem

Let r = x i + y j + z k and r = |r|.32. If F = r/rp, find div F. Is there a value of p for which div F = 0?

To determine

To find: The divF and value of p, when divF=0 .

Explanation

Given data:

r=xi+yj+zk

Formula used:

Consider the standard equation of a divergence of vector field.

divF=Px+Qy+Rz (1)

Find r .

r=|r|=|xi+yj+zk|=x2+y2+z2

Find F=rrp .

F=rrp

Substitute xi+yj+zk for r and x2+y2+z2 for r ,

F=xi+yj+zk(x2+y2+z2)p=x(x2+y2+z2)pi+y(x2+y2+z2)pj+z(x2+y2+z2)pk=x(x2+y2+z2)p2i+y(x2+y2+z2)p2j+z(x2+y2+z2)p2k

Substitute x(x2+y2+z2)p2 for P , y(x2+y2+z2)p2 for Q and z(x2+y2+z2)p2 for R in equation (1),

divF=x[x(x2+y2+z2)p2]+y[y(x2+y2+z2)p2]+z[z(x2+y2+z2)p2]={[(x2+y2+z2)p2(x)p2(x2+y2+z2)p21(2x)((x2+y2+z2)p2)2]+[(x2+y2+z2)p2(y)p2(x2+y2+z2)p21(2y)((x2+y2+z2)p2)2]+[(x2+y2+z2)p2(z)p2(x2+y2+z2)p21(2z)((x2+y2+z2)p2)2]} {t(uv)=vt(u)ut(v)v2}

Simplify the equation.

divF={[(x2+y2+z2)p2(x2)p(x2+y2+z2)p21(x2+y2+z2)p]+[(x2+y2+z2)p2(y2)p(x2+y2+z2)p21(x2+y2+z2)p]+[(x2+y2+z2)p2(z2)p(x2+y2+z2)p21(x2+y2+z2)p]}={(x2+y2+z2)p2[1(x2)p(x2+y2

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