Prove the vector area formula using line and surface integrals

[rbwang1225]

I met a proof problem that is as follows.
$\bf a = ∫_S d \bf a$, where S is the surface and $\bf a$is the vector area of it.
Please proof that $\bf a = \frac{1}2\oint \! \bf r \times d\bf l$, where integration is around the boundary line.

Any help would be very appreciated!

 

[GarageDweller]

Strokes theorem?

hmmm

Well say you perform a surface integral, if the vector field in question is the normal vector of the surface, then the only thing left in the integrand is dA (scalar).

So I guess using strokes theorem, you have to find a vector field who's curl is the normal vector of the surface.

 

[DonAntonio]

Strokes theorem?

hmmm

Well say you perform a surface integral, if the vector field in question is the normal vector of the surface, then the only thing left in the integrand is dA (scalar).

So I guess using strokes theorem, you have to find a vector field who's curl is the normal vector of the surface.

Strokes Theorem is what we get sometimes from our loved ones. Stokes Theorem is, perhaps, what you mean.

DonAntonio

 

[GarageDweller]

Oops

 

[chiro]

Strokes Theorem is what we get sometimes from our loved ones. Stokes Theorem is, perhaps, what you mean.

DonAntonio

What's this new theorem and who proved it lover boy?