Surface integral to a volume integral

[esander4]

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.

 

[Deleted member 4993]

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.

Do you remember:

Green's theorem?

Stoke's theorem?

 

[esander4]

Do you remember:

Green's theorem?

Stoke's theorem?

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.

 

[Deleted member 4993]

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.

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...

 

[renegade05]

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...

Calc IV actually.

 

[esander4]

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...

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

 

[esander4]

Still struggling to find the answer. Would Divergence theorem be closer to what I need?

 

[Deleted member 4993]

Still struggling to find the answer. Would Divergence theorem be closer to what I need?

Yes ... and it is also sometimes known as Ostrogradsky's theorem. It is a special case of Stoke's theorem.

 

[esander4]

Yes ... and it is also sometimes known as Ostrogradsky's theorem. It is a special case of Stoke's theorem.

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