Question

Use the Divergence Theorem to compute the net outward flux of the following field across the given surface S.

F = < 8y^2 - 3x, -9x+4y, -2y^3 +z >

S is the sphere {(x,y,z): x^2 + y^2 + z^2 = 9}

Find net outward flux across the surface.

Answer 1

Answer:

Flux across S = 72π

Step-by-step explanation:

First we need to calculate the divergence of the vector field:

Div F = $\frac{dFx}{dx} + \frac{dfy}{dy} + \frac{dFz}{dz}$

Where

Fx = 8y^2 - 3x

Fy = -9x+4y

Fz = -2y^3 +z

Then

Div F = -3 +4 + 1  = 2

And how the vector field’s divergence is a constant, we can calculate the flux across of the surface how:

Flux across S = Div F * Volume of Sphere

Fluz acroos S = 2(4/3)π$r^{3}$

                 r : Sphere’s radio

Flux across S = (2)(4/3)π$3^{3}$

                       = 72π