We are not going to do that work for you, but will be glad to help you.

The unit sphere is the sphere with radius 1 centered in the origin. They give you the coordinates of a vector. Compute the dot product of the vector and the local surface area normal and integrate it over the spherical surface area. Then do it again with Gauss' theorem, which involves a volume integral of the divergence of the vector. They should be equal.

We will be glad to critique your work. There is quite a bit involved.

Compute the flux of the vector field F = (3xy2; 3x2y; z3) out of the unit sphere both
directly and using Gauss' Theorem.

Show all work please!!!!!!

Direct computation:

The outward normal on the unit sphere is, of course:

n = (x,y,z)

Take inner product with F:

F dot n =

(3xy^2, 3x^2y, z^3) dot (x,y,z) =

6 x^2y^2 + z^4

Now switch to spherical coordinates:

x = sin(theta) cos(phi)

y = sin(theta) sin(phi)

z = cos(theta)

F dot n =

6 sin^4(theta) cos^2(phi)sin^2(phi)+ cos^4(theta)

Surface element on unit sphere in spherical coordinates is:

sin(theta) d theta d phi

So we have to integrate:

[6 sin^4(theta) cos^2(phi)sin^2(phi)+ cos^4(theta)] sin(theta) d theta d phi

from phi = 0 to 2 pi and theta = 0 to pi.

The integrals are high school level, the result is:

12 pi/5

Now using Gauss Theorem, compute the divergence of F:

div F = d(3xy^2)/dx + d(3x^2y)/dy +
d(z^3)/dz = 3(x^2 + y^2 + z^2) = 3 r^2

We must integrate this over the volume of the unit sphere. The integrand does not depend on the angles so, we can write the integral as:

Integral from 0 to 1 of
3 r^2* 4 pi r^2 dr

= 12 pi/5

To compute the flux of the vector field F = (3xy^2, 3x^2y, z^3) out of the unit sphere using direct computation, we need to find the dot product of the vector field with the outward normal on the unit sphere.

1. Find the dot product:
F dot n = (3xy^2, 3x^2y, z^3) dot (x,y,z)
= 3xy^2 * x + 3x^2y * y + z^3 * z
= 3x^2y^2 + z^4

2. Switch to spherical coordinates:
x = sin(theta) * cos(phi)
y = sin(theta) * sin(phi)
z = cos(theta)

3. Rewrite the dot product in terms of spherical coordinates:
F dot n = 3(sin(theta) * cos(phi))^2 * (sin(theta) * sin(phi))^2 + cos(theta)^4
= 3sin^4(theta) * cos^2(phi) * sin^2(phi) + cos^4(theta)

4. Determine the surface element on the unit sphere in spherical coordinates:
The surface element on the unit sphere in spherical coordinates is sin(theta) * d(theta) * d(phi).

5. Integrate the dot product over the spherical surface area:
We need to integrate [3sin^4(theta) * cos^2(phi) * sin^2(phi) + cos^4(theta)] * sin(theta) * d(theta) * d(phi) from phi = 0 to 2pi and theta = 0 to pi.

6. Perform the integrals:
Integrating with respect to phi:
∫[0 to 2pi] cos^4(theta) d(phi) = 2pi

Integrating with respect to theta:
∫[0 to pi] 3sin^4(theta) * cos^2(phi) * sin^2(phi) + cos^4(theta) d(theta) = 4pi/5

7. Multiply the results together:
12pi/5

So, the flux of the vector field F out of the unit sphere using direct computation is 12pi/5.

Now, let's use Gauss' Theorem to compute the flux:

1. Compute the divergence of F:
div F = d(3xy^2)/dx + d(3x^2y)/dy + d(z^3)/dz
= 3(x^2 + y^2 + z^2)
= 3r^2

2. Integrate the divergence over the volume of the unit sphere:
We need to integrate 3r^2 * 4pi * r^2 dr from r = 0 to r = 1.

3. Perform the integral:
∫[0 to 1] 3r^4 * 4pi dr = 12pi/5

So, the flux of the vector field F out of the unit sphere using Gauss' Theorem is also 12pi/5.

Therefore, the flux of the vector field F out of the unit sphere is 12pi/5, both directly computed and using Gauss' Theorem.