What is gained by vector calculus?

[hobz]

I am currently reading about the divergence and I am curious what is gained by reformulating stuff like Gauss law from integral form into differential form?

 

The expressions become shorter and nicer, but surely there must be another reason?

[ajb]

The biggest advantage I can see writing things in differential form is that one can pass to differential forms which is not only very compact notationally but allows for many generalisations.

[Bignose]

In the most general of terms, sometimes it is easier to know how the quantity (of whatever you are measuring) changes in a volume, and sometimes it is easier to know the fluxes going into and out of the surface of that volume, and it is really darn nice to know how to relate the two.

[hobz]

In the most general of terms, sometimes it is easier to know how the quantity (of whatever you are measuring) changes in a volume, and sometimes it is easier to know the fluxes going into and out of the surface of that volume, and it is really darn nice to know how to relate the two.

 

Could you give a practical example of this?

[ajb]

Although it is from a pure mathematics point of view, the pedagogical paper "A Tasty Combination: Multivariable Calculus and Differential Forms" by Edray Herber Goins and Talitha M. Washington may be of interest.

 

It goes as far as the generalised Stoke's theorem but just stops short of de Rham cohomology.

 

arXiv:0910.0047

[Bignose]

Could you give a practical example of this?

 

Most derivations of the fluid mechanics and other conservation equations will use the divergence theorem.

 

http://en.wikipedia.org/wiki/Derivation_of_the_Navier%E2%80%93Stokes_equations

 

The divergence theorem is used up there near the top.

[hobz]

I see. I will give both a look. Thanks!

[D H]

Most derivations of the fluid mechanics and other conservation equations will use the divergence theorem.

Flip side: Most simulations of fluid dynamics (e.g., Computational fluid dynamics) and other conservation equations (e.g. galaxy simulations) use the differential equations -- and they try to use integration techniques that do a reasonably good job of ensuring that energy is conserved.

[Shadow]

Not that I have much of an idea what this thread is about, but "reasonably" made me smile