Calculus on Manifolds: Meaning & Benefits | Mechanical Engineer

[impelluso]

Hello,

I am a mechanical engineer and I am teaching my self the topic of this subject line.

I now have a working understanding of the following: manifolds, exterior algebra, wedge product and some other issues. (I give you this and the next sentence so I can CONTEXTUALIZE my question.) I can appreciate that I am just a few hours of work away from understanding Stoke's Theorem, and the hierarchy of gradient, curl, diverence and how they are all Cartesian specifications of forms and so on.

But I am at an obstacle that is confounding me...

What does this sentence MEAN: Doing Calculus on Manifolds.

How does one DO calculus on manifolds?
WHY IS USEFUL to do calculus on manifolds?
Is it nothing more than developing differential equations in alternate coordinate systems?

Keep in mind that I am steeped in the old way of doing things and am likely blind to the obvious.

 

[Office_Shredder]

Doing calculus on manifolds just means having a definition of integration and differentiation on a manifold (as opposed to just R^k), and being able to calculate those integrals and derivatives.

The reason it's useful is that we often want to calculate integrals and derivatives on things that are not R^k (such as a surface in R³ where we use Stoke's theorem).