Advanced Calculus

[bigbadwolf]

What are some advanced calculus texts people here like?

 

[IntoDarkness]

wat is "advanced calculus"? ode? matrix calculus? complex variable? analysis?

 

[bigbadwolf]

wat is "advanced calculus"? ode? matrix calculus? complex variable? analysis?

There isn't complete consensus on what topics should be included. Multivariable differentiation and integration for sure. Vector analysis (with or without differential forms) up to Stokes' theorem. In the older books there might be a single chapter for each of complex variables and ODEs. Also the treatment would usually be rigorous (epsilons and deltas). The problem is that in the early '70s the standard one-year course in "advanced calculus", as taught to mathematicians, physicists, and engineers, was done away with and replaced with the more fragmented courses of today.

 

[Daniel Duffy]

Maybe 'calculus' is a common name in USA. In Europe calculus is what we learn(ed() at school. More accurate names imo are math methods (calculus on steroids, working things out like Stokes stuff) and real/complex analysis (epsilon/delta, hard core stuf). There again, I haven't looked in 47 years.

School of Mathematics - Trinity College Dublin

maths.tcd.ie

 

[Daniel Duffy]

From AMS Classification, 'calculus' on its own is ambiguous, so many 'calculusses' to choose from.

MSC2010 database

mathscinet.ams.org

 

[Daniel Duffy]

On a related issue when someone says "I know linear algebra and calculus", it could mean almost anything.

 

[WZhang]

This is a very ambigious definition as advance calculus goes. It can be anything Calculus 3 (Strokes,greens,curls, and parameterisation). Is it ODE? We have no idea what you are describing here.

 

[bigbadwolf]

On a related issue when someone says "I know linear algebra and calculus", it could mean almost anything.

What books other than the one by Widder would you recommend for advanced calculus?

 

[Daniel Duffy]

What books other than the one by Widder would you recommend for advanced calculus?

It depends on the target/reader group. I work backwards to the books that satisfy the needs.

I still don't know the precise definition of "Advanced Calculus"..

 

[Daniel Duffy]

I would say sections B, C, D, E correspond subsume calculus

https://www.datasim.nl/application/files/1315/5947/7165/DL_Pure_Mathematics_Foundations.pdf

Calculus is not analysis; the latter teaches differentiability while the former just 'does it'.

Calculus == HOW
Analysis == WHEN,WHY,WHAT IF,WHAT KINDS OF

 

[Qui-Gon]

Sorry to revive a somewhat old thread.

At least in the U.S., advanced calculus tends to act as a bridge between traditional multivariable calculus — calculus of scalar valued functions defined on R^2, R^3 — and differential geometry. That is, advanced calculus tends to function as a true vector calculus course; in my experience, multivariable calculus didn’t spend enough time treating the calculus of vector-valued maps. Spivak has a great little book titled “Calculus on Manifolds”; Marsden and Tromba’s “Vector Calculus” is also terrific; finally, Loomis and Sternberg’s “Advanced Calculus” treats some even more advanced topics like Lie derivatives and Exterior calculus.

In the States, the main difference between calculus and analysis is that the former is calculation-based and the latter is proof-based.

 

[Daniel Duffy]

Mathematical Analysis is more than proving theorems,(which are neither necessary nor sufficient to say that you are dong analysis).

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.[1][2]

Historically,

The modern foundations of mathematical analysis were established in 17th century Europe.[3] Descartes and Fermat independently developed analytic geometry, and a few decades later Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete problems by continuous ones.

Nunerical Analysis + Real Analysis is a powerful cocktail.

 

[Qui-Gon]

I should have been more careful. I was meaning to say the main differences in *how calculus and analysis are taught*; I was not meaning to make any claim about what the two meant historically. In this sense, analysis (real, complex, Fourier, and functional) is most certainly about proving theorems. At the undergraduate level, numerical analysis doesn’t seem to be a proper subcamp of analysis, though this could simply be because of how it is taught at my university where only calculus, linear algebra and ODEs are prerequisite courses.

To stop with Newton and Leibniz is an injustice, however. Cauchy and Weierstrass and others of the 19th century are equally worth mention in the development of modern analysis. In my opinion, the 1800's is when analysis (using this as a pure mathematician would) got underway.

My initial response was simply to indicate that “advanced calculus”, as a university level course, typically functions as a bridge to learning differential geometry and treating vector calculus techniques more generally.

 

[Quasar Chunawala]

For vector calculus, someone on the QN forum had suggested, Colley S.J. and I ended up working through the first four chapters of the book. It's a really nice book, the selection of examples is cool!