Use the divergence theorem to compute the net outward flux of the vector field f across the boundary of region d. d is the region between the spheres of radius 4 and 5 centered at the origin. f = < 9z+4x, x-7y, y+9z>

The net outward flux of the vector field F across the boundary of region D is 488 and this can be determined by using the divergence theorem.

Given :

D is the region between the spheres of radius 4 and 5 centered at the origin. F = <9z+4x, x-7y, y+9z>

According to the divergence theorem:

Now, the expression for is given by:

Now, the spherical coordinates is given by:

x = u cosv sinw

y = u sinv sinw

z = u sinw

Therefore, the value of dV is given by:

Now, the net outward flux of the vector field F across the boundary of region D is given by:

Simplify the above integral.

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By the divergence theorem,

We have

In the integral, convert to spherical coordinates, taking

so that

Then the flux is

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