Surface integral to a volume integral

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esander4

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I'm having a hard time remembering how to do this from my last calculus class (this is for a Materials Kinetics and Transport Phenomena class, doing a math review).

Convert the following surface integral to a volume integral

\(\displaystyle \iint(\bigtriangledown\times\underset{V}{\rightharpoonup})\cdot (\underset{n}{\rightharpoonup})dA\)

I know how to do cross products and dot products, but I'm kind of confused on it being such a general solution.
 
esander4 said:
I'm having a hard time remembering how to do this from my last calculus class (this is for a Materials Kinetics and Transport Phenomena class, doing a math review).

Convert the following surface integral to a volume integral

\(\displaystyle \iint(\bigtriangledown\times\underset{V}{\rightharpoonup})\cdot (\underset{n}{\rightharpoonup})dA\)

I know how to do cross products and dot products, but I'm kind of confused on it being such a general solution.
Click to expand...

Do you remember:

Green's theorem?

Stoke's theorem?
 
Subhotosh Khan said:
Do you remember:

Green's theorem?

Stoke's theorem?
Click to expand...

I remember the concepts, like Green's theorem being from a line integral to surface and Stoke's being a more general form of Green's theorem, but the specifics have been lost to me.
 
esander4 said:
I remember the concepts, like Green's theorem being from a line integral to surface and Stoke's being a more general form of Green's theorem, but the specifics have been lost to me.
Click to expand...

To find the answer to your question (since you are doing a review), those specifics must be recovered. Time to dust up those old calc-III books...
 
Subhotosh Khan said:
To find the answer to your question (since you are doing a review), those specifics must be recovered. Time to dust up those old calc-III books...
Click to expand...

So far all I've been able to come up with is that it transforms to

\(\displaystyle \iint(\partial P/\partial x-\partial Q/\partial y)dA\)

but I can't find anything about transforming it to a volume integral
 
Still struggling to find the answer. Would Divergence theorem be closer to what I need?
 
esander4 said:
Still struggling to find the answer. Would Divergence theorem be closer to what I need?
Click to expand...

Yes ... and it is also sometimes known as Ostrogradsky's theorem. It is a special case of Stoke's theorem.
 
Subhotosh Khan said:
Yes ... and it is also sometimes known as Ostrogradsky's theorem. It is a special case of Stoke's theorem.
Click to expand...

This problem seems to be still eluding me. So far I know that

\(\displaystyle \iiint(\bigtriangledown\cdot v))dA\) goes to \(\displaystyle \iint(v\cdot n))dA\) and the other way around but I can't find anything about converting a scalar triple product surface integral to volume integral
 
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