What Comes After Vector Calculus in Pure Calculus?

[inversquare]

Are there fields of "pure" Calculus that follow Vector Calculus? I mean fields that are not primarily ODE, PDE etc, Real Analysis or Complex Analysis, topology etc. Does Vector Calculus and its prerequisites fully encompass the basics of Calculus as a field?

 

[jedishrfu]

there's tensor analysis and differential forms.

 

[inversquare]

Concise, thank you jedishrfu. May the force be with you.

 

[jedishrfu]

It was particularly strong two days ago :-)

 

[micromass]

Calculus of variations might qualify.

And of course, there are many calculus subjects that can be explored more indepth than what is usually seen in calculus courses. For example, there are other techniques of solving integrals than the ones usually seen (ie substitution and by parts). One that comes to mind immediately is differentiation under the integral sign.

 

[inversquare]

I see micromass. Would you consider the Residue Theory from Complex Analysis as applied to real integrals to fit this description as well?

 

[micromass]

I see micromass. Would you consider the Residue Theory from Complex Analysis as applied to real integrals to fit this description as well?

Perhaps. But I think it has a very different flavour than usual calculus. So I wouldn't consider it part of calculus.

 

[jedishrfu]

And Calculus of Variations...

 

[SteamKing]

I see micromass. Would you consider the Residue Theory from Complex Analysis as applied to real integrals to fit this description as well?

Perhaps. But I think it has a very different flavour than usual calculus. So I wouldn't consider it part of calculus.

Why not? The Residue Theorem has got it all: integrals, infinite series, limits, Stokes' Theorem, etc. Seems like those topics are "usual" calculus.

 

[micromass]

I guess it's a matter of taste. I understand why you would include them in usual calculus, but I don't feel it that way.

 

[Chestermiller]

Would vector calculus also include differential geometry?

 

[FactChecker]

I see micromass. Would you consider the Residue Theory from Complex Analysis as applied to real integrals to fit this description as well?

You might get better answers by specifying areas of interest and applications than by trying to classify subjects this way.

 

[inversquare]

You might get better answers by specifying areas of interest and applications than by trying to classify subjects this way.

Just trying to get an idea of what micromass was talking about here:

there are other techniques of solving integrals than the ones usually seen

From my perspective it seems as though the edges of "pure" calculus get a bit fringy as they head off into other fields; some applications are not clear cut as one discipline or another.

 

[mathwonk]

just restricting to differential calculus, this is the theory of approximating non linear functions locally by linear ones. the term vector calculus' usually means to me this theory carried out in a finite number of dimensions. if this is your experience then the enxt stage woulod be infinite dimensional (banach space) differential calculus,. which also encompasses calculus of variations.

then there is integral calculus, which also can be done in finite dimensionalm vector spaces or on m anifolds, or in general measure spaces.

 

[inversquare]

just restricting to differential calculus, this is the theory of approximating non linear functions locally by linear ones

Interesting. This is how you would generally define differential calculus?

 

[hunt_mat]

There are Sobolev spaces, measure theory and something which I have been involved in called geometric algebra and geometric calculus.

 

[phion]

Complex analysis is, in my humble opinion, one of the coolest, and beautiful branches of mathematics that subsequently requires the entire calculus sequence to fully comprehend. It's a tremendously powerful and useful mathematics with many applications relevant to calculus inside pure math as well as physics and engineering.

 

[hunt_mat]

It is however constrained to two dimensions only.

 

[micromass]

There is complex analysis in multiple variables too.

 

[hunt_mat]

Quaternion analysis? Or several complex variables?

 

[mathwonk]

In hormander's little book on several complex variables he does essentially all of the significant results of one complex variable in chapter one, some 20-30 pages long, (including the cauchy integral theorem for smooth but not necessarily analytic functions), prompting my several complex variables teacher (Hugo Rossi) to remark, arguably of course, "he shows you what a mickey mouse subject one complex variable really is!".

https://www.amazon.com/dp/0444884467/?tag=pfamazon01-20

 

[lavinia]

Would vector calculus also include differential geometry?

In my opinion, no. Calculus is necessary for Differential Geometry but it is also necessary for a lot of mathematics. Differential Geometry deals with connections on vector bundles. To me this is another topic.