Verifying Divergence Theorem on Sphere with F(x,y,z)=zi+yj+xk

[bugatti79]

Homework Statement

Folks,

Verify the divergence theorem for

F(x,y,z)=zi+yj+xk and G the solid sphere x^2+y^2+z^2<=16

Homework Equations

$\int\int\int div(F)dV$

The Attempt at a Solution

My attempt

The radius of the sphere is 4 and div F= 1, therefore the integral becomes

$\int\int\int div(F)dV=\int_0^{2\pi} \int_0^{\pi} \int_0^{4} 1dV=\int_0^{2\pi} \int_0^{\pi} \int_0^{4} 2\rho^2 sin (\phi) d\rho d\phi d \theta$

Is this correct so far?
Thanks

 

[cjc0117]

where did the 2 come from in the integrand? Other than that it's right.

EDIT: Also, since divF is a constant and the volume of a sphere can easily be found, you don't actually need to evaluate an integral.

 

[bugatti79]

where did the 2 come from in the integrand? Other than that it's right.

EDIT: Also, since divF is a constant and the volume of a sphere can easily be found, you don't actually need to evaluate an integral.

Hi, that should be a 1, ie

$...=\int_0^{2\pi} \int_0^{\pi} \int_0^{4} \rho^2 sin (\phi) d\rho d\phi d \theta$
But it says I need to verify the Divergence Theorem...so I guess I can continue and verify?

Thanks

 

[DryRun]

\int_0^{2\pi} \int_0^{\pi} \int_0^{4} dV = volume of sphere with radius 4.

Just use the well-known formula for calculating volume of sphere. The answer should obviously be the same as the evaluation of your triple integral expression above.

Now, when you're asked to verify the Gauss Divergence Theorem, you should independently calculate the total flux using the surface integral formula, to confirm your answer.
\iint_S \vec F \hat n \,.d\sigma
Hint: To make it easier, split the sphere exactly in half laterally and evaluate the surface integral for one half. Then multiply by 2 to get the total flux.

 

[bugatti79]

\int_0^{2\pi} \int_0^{\pi} \int_0^{4} dV = volume of sphere with radius 4.

Just use the well-known formula for calculating volume of sphere. The answer should obviously be the same as the evaluation of your triple integral expression above.

Now, when you're asked to verify the Gauss Divergence Theorem, you should independently calculate the total flux using the surface integral formula, to confirm your answer.
\iint_S \vec F \hat n \,.d\sigma
Hint: To make it easier, split the sphere exactly in half laterally and evaluate the surface integral for one half. Then multiply by 2 to get the total flux.

Ok, thanks guys. Will respond hopefully at some stage.

Cheers

 

[DryRun]

Note: d\sigma is the differential area.