Quick Divergence Theorem question

[Stef42]

Homework Statement

Use the divergence theorem in three dimensions

$$\int\int\int\nabla\bullet V d\tau= \int \int V \bullet n d \sigma$$

to evaluate the flux of the vector field

V= (3x-2y)i + x⁴zj + (1-2z)k

through the hemisphere bounded by the spherical surface x²+y²+z²=a² (for z>0) and the x-y plane

Hint: The direct evauation of the flux may not be the easiest way to proceed

Homework Equations

The Attempt at a Solution

I found it pretty simple which means I probably messed up (and I'm not sure what the hint is talking about
ok so the divergence is
$$\nabla \bullet V = 3-2=1$$

and the integral over the volume of the hemisphere (using spherical polar coordinates) is
$$\int_{0}^a r^2 d \tau \int_{0}^{2\pi} d\phi \int_{0}^{\pi/2} sin\theta d\theta = \frac{2\pi a^3}{3}$$

So am I doing it completely wrong? I don't know the answer but if anyone could look through it and spot anything I would really appreciate it

 

[tiny-tim]

Hi Stef42!

(have a del: ∇ and a theta: θ and a phi: φ and a sigma: σ and use \cdot instead of \bullet )

Use the divergence theorem in three dimensions

$$\int\int\int\nabla\bullet V d\tau= \int \int V \bullet n d \sigma$$

to evaluate the flux of the vector field
…
Hint: The direct evauation of the flux may not be the easiest way to proceed
…
I found it pretty simple which means I probably messed up (and I'm not sure what the hint is talking about

Looks fine to me …

I think the hint just means don't use V.n, which is the flux, use ∇.V … though that's a bit unnecessary since they've already told you to

 

[Stef42]

thanks for the speedy response tiny-tim :-)
Ok so just to clear up:
∇.V= divergence of vector field V
V.n= flux
So then the sigma integral is flux density ?
So what did I calculate? flux density aswell? wasn't I supposed to find the flux?

 

[tiny-tim]

So what did I calculate? flux density aswell? wasn't I supposed to find the flux?

oooh, I get confused between flux and flux density …

anyway, V.n is definitely fluxy, and ∇.V definitely isn't

 

[Stef42]

well, at least I'm not the only one who gets confused, damn vector calculus
well from my lectures notes:
"The divergence represents the flux density of the vector field and, because of the derivative operation, has an associated Fundamental Theorem of Calculus called the divergence theorem : (quotes equation)"

hmm think I'll go talk to him tomorrow :)