Fundamental theorem of calculus for surface integrals?

[Jhenrique]

Hellow!

A simple question: if exist the fundamental theorem of calculus for line integrals not should exist too a fundamental theorem of calculus for surface integrals? I was searching about in google but I found nothing... What do you think? Such theorem make sense?

 

[Boorglar]

Yes, there are similar theorems for surface integrals. Look up Stokes' Theorem, and also the Divergence Theorem.

Stokes' theorem says that the surface integral of the curl of a vector field on a surface in R^3 is equal to the line integral of the vector field on the boundary of the surface.

The Divergence theorem says that the triple integral of the divergence of a vector field in a volume in R^3 is equal to the flux of the vector field through the surface which bounds the volume.

http://en.wikipedia.org/wiki/Kelvin–Stokes_theorem
http://en.wikipedia.org/wiki/Divergence_theorem

 

[Jhenrique]

I disagree. The divergence theorem connects a double integral of a closed surface with a triple integral, similarly, the rotational theorem connects a simple integral of a closed curve with a dobule integral. These concepts are different from the fundamental theorem of calculus...

 

[Number Nine]

I disagree. The divergence theorem connects a double integral of a closed surface with a triple integral, similarly, the rotational theorem connects a simple integral of a closed curve with a dobule integral. These concepts are different from the fundamental theorem of calculus...

Actually, all are just special cases of Stokes' theorem. They are deeply related.

 

[Jhenrique]

I disagree again. The divergence theorem is used to calculate a volume by means of a triple integral or by means of a double integral of a closed surface of this volume. The rotational theorem is used to calculate an area by means of a double integral or by means of a simple integral of a closed curve of this area.
The fundamental theorem of calculus for line integrals is a concept that relates to the independence of a path, similarly, the fundamental theorem of calculus for surface integrals should give an explanation for integrals independent of the surface of integration.

 

[Number Nine]

I disagree again. The divergence theorem is used to calculate a volume by means of a triple integral or by means of a double integral of a closed surface of this volume. The rotational theorem is used to calculate an area by means of a double integral or by means of a simple integral of a closed curve of this area.
The fundamental theorem of calculus for line integrals is a concept that relates to the independence of a path, similarly, the fundamental theorem of calculus for surface integrals should give an explanation for integrals independent of the surface of integration.

They are, in fact, all just special cases of Stokes' theorem (i.e. they all follow immediately). If you want "independence of surfaces", let F be a C¹ vector field and let S₁ and S₂ be surfaces with a common boundary B (with all of the usual assumptions). By the Kelvin-Stokes theorem, the surface integrals of the curl of F over S₁ and S₂ are equal to the line integral of F over B, and so are identical. The value of the surface integral depends only on the line integral around the boundary.

 

[Jhenrique]

The rotational theorem says:

The circulation of a vector field over a closed path C is equal to the integral of the normal component of the curl of that field over a surface S for which C is a boundary.

or:
$$\oint \vec{f}\cdot d\vec{s}=\iint |\vec{\bigtriangledown}\times\vec{f}|\;\;dxdy$$

And the divergence theorem says:

The flux of a vector field through a closed surface S is equal to the integral of the divergence of that field over a volume V for which S is a boundary

or:
$$\iint\!\!\!\!\!\!\!\!\!\!\subset\!\supset \vec{f}\cdot d\vec{S}=\iiint |\vec{\bigtriangledown}\cdot \vec{f}|\;\;dxdydz$$

* Of course the two circulations below are equals, independent of the path of integration. And the same goes for the two fluxes below, they are equals, independent of the surface of integration.

But now I ask you. What all this have to do with the fundamental theorem of calculus?

 

[Number Nine]

What all this have to do with the fundamental theorem of calculus?

They are all special cases of Stokes' theorem, as I said. Stokes' theorem unifies not only the fundamental theorem of calculus, but almost every major result in vector calculus. The link above explains everything quite well. Stokes' theorem relates integrals over the boundary of a manifold (a generalization of a surface) to integrals over the manifold itself. A curve is a one dimensional manifold, and its boundary consists of points; Stokes' theorem implies the fundamental theorem of calculus/line integrals in this case. The boundary of a 2 dimensional manifold (a surface) is a curve; Stokes' theorem implies most of the results from vector calculus in this case.

 

[Jhenrique]

Hmmm, now I undertood, so so, because I don't know how make a exterior derivative. But, anyway, exist a geometric explanation for the fundamental theorem of calculus for surface integral, a geometric explanation analogous to t.f.c. for line integral?

 

[ibdsm]

I disagree. The divergence theorem connects a double integral of a closed surface with a triple integral, similarly, the rotational theorem connects a simple integral of a closed curve with a dobule integral. These concepts are different from the fundamental theorem of calculus...

Stokes Theorem connects (n+1)-fold integrals over a region with n-fold integrals over the boundary of the region.

The fundamental theorem of calculus is the case n=0. (An integral over a discrete set is a sum - use counting measure)