   Chapter 16, Problem 36RE

Chapter
Section
Textbook Problem

Compute the outward flux of F(x, y, z) = x i+ y j+ z k ( x 2 +   y 2 +   z 2 ) 3 2 through the ellipsoid 4x2 + 9y2 + 6z2 = 36.

To determine

To compute : The outward flux of F(x,y,z)=xi+yj+zk(x2+y2+z2)32 through the ellipsoid 4x2+9y2+6z2=36 .

Explanation

Given data:

The vector field is F(x,y,z)=xi+yj+zk(x2+y2+z2)32 through the ellipsoid 4x2+9y2+6z2=36 .

Formula used:

Write the expression Divergence Theorem.

EdivFdV=S1F(n1)dS+S2F(n2)dS (1)

Here,

E is the solid region.

Write the expression for surface area of the sphere.

S1dS=4πr2 (2)

Consider the value r as follows.

r=xi+yj+zk

Find r .

r=(xi+yj+zk)(xi+yj+zk)=xxii+yyjj+zzkk=xx(1)+yy(1)+zz(1) {ii=1,jj=1,kk}=xx+yy+zz

Simplify expression as follows.

r=1+1+1=3

Find r .

r=(xi+yj+zk)(|xi+yj+zk|)=(xi+yj+zk)(x2+y2+z2)=x(x2+y2+z2)i+yy(x2+y2+z2)j+zz(x2+y2+z2)k=[12(x2+y2+z2)12(2x)]i+[12(x2+y2+z2)12(2y)]j+[12(x2+y2+z2)12(2z)]k

Simplify the equation as follows.

r=(xx2+y2+z2)i+(yx2+y2+z2)j+(zx2+y2+z2)k=1x2+y2+z2(xi+yj+zk)

r=r|r| (3)

Consider S1 be the sphere of radius 1 with center at the origin and outer unit normal n1 . Consider S2 be the ellipsoid surface with outer unit normal n2 and the region between S1 and S2 is represented as E

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