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I am a first-year graduate student in mathematics. My undergraduate mathematics curriculum did not emphasize "calculating"; it was a theoretical curriculum in which even a traditional course in multivariable calculus was not "required" (a course in differential geometry sufficed).

I am training to be a "hands-on analyst", if that term makes any sense. For example, I know how to existence and uniqueness of solutions to PDE, but I haven't yet the "nose" to compute, to perform certain critical integration by parts, etc. I am starting to realize that theories are built on calculations and certain very interesting techniques in PDE--such as viscosity methods for example--arose from refining one's intuition while performing calculations. This is very inspiring for me and I want to learn to calculate!

Calculating has been an acquired taste for me, and as a "hands-on analyst", I would like to work in PDE and variational problems where one is interested in producing sharp bounds, etc. (this is vague, I know).

I am wondering if anyone can suggest any references/ workbooks where I can refine my "computation" skills. For example, I heard that the physicist Lev Landau gave his prospective students a preliminary test in integration. I suspect I will not pass such a test at this moment, but I would like to try to get myself to a stage where I can. Is there perhaps (a Russian?) text that emphasizes computation and serves as a good workbook for refining one's computation/calculation abilities.

Much thanks in advance!

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You might find Schaum's 3,000 Solved Problems in Calculus a good review of calculus. It is mostly computational and the answers/explanations are given right after each problems. – Student Jan 31 '13 at 1:16
I really like this question @user – Rustyn Jan 31 '13 at 1:23
Are you looking for applied books in PDE's, Numerical Methods and calculating in general? Regards – Amzoti Jan 31 '13 at 4:07
I think V.I. Arnold's Trivium (and it's follow-up) might be useful to you. – Potato Aug 4 '13 at 18:10

As suggested in the comments above: Schaum's 3,000 Solved Problems in Calculus seems to fit the bill! You can preview the table of contents and some sample pages at the link above.

Another such collection is entitled The Humongous Book of Calculus Problems by Michael Kelley, relatively inexpensive (as is the Schaum's book), and here, to, the link will take you to where you can preview the book.

Also Khan's Academy Calculus has great videos for review, and includes practice questions to work through.

In addition, you might want to check out Paul's Online Math Notes, click on the drop down menu for "class notes" and you'll find tutorials with practice problems for Calc I, II, and III.

Finally, this site: Math.SE, has loads of posted questions (many, many computational in nature) related to Calculus (and derivatives, integrals, etc), and most questions have one or more answers/hints to solutions. And if you find a problem somewhere that you can't seem to solve, your welcome to ask it here! (And you're welcome to use your refreshed, developing computational skills by answering questions, as well!)

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+1 Reading Paul's Online Math Notes (and doing the exercises) was literally the way I learned multivariable calculus (Calc III). – Jesse Madnick Jan 31 '13 at 3:27
+1 for nice leading. Thanks for sharing it. – Babak S. Jan 31 '13 at 6:47

Here's another perspective: Study some physics! Physics problems mostly deal with computing things and I can imagine you'd be able to pick up things very fast with a math background. And you won't be learning just calculations but you'll also be learning some very interesting things. If you pick up any book on Classical or Quantum Mechanics, or E&M, most involve interesting math and the problems involve lots of computations and tricks involved for those computations.

If you can do the problems in Stone and Goldbart's "Mathematics for Physics" book, I'd say your computation skills are pretty solid.

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Polya & Szego "Problems and theorems in analysis", also translated to Russian.

These two volumes are full of problems that teach you to calculate, but it is also full of ideas and the extremely useful analytic folklore, which may be unknown to modern students. Everything is developed through series of problems, so it is a major undertaking to do the whole thing, but it is extremely rewarding. The only bad thing about it is that its scope is mostly limited to 19th century single-variable function theory.

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I'm in a similar position to you, insofar as I finished a math degree about 10 years ago, and I'm now preparing to switch careers and am preparing for professional exams. I lost all my chops.

I've been using Calculus on the Web to strengthen those skills. Temple University's math department uses it to automate homework, and it is open to the public. It's pretty neat -- they break down problems into the essential steps in the first few problem sets on a topic, so you drill the procedure to do it right.

Also, I strongly suggest getting your hands on a flashcarding program like Anki and memorizing any derivatives or integrals (and any other computed values) you commonly see. Being good at "hands on" analysis means being able to use experience and knowledge to solve problems quickly. You can "fake" experience with knowledge. In short, you want the "common" facts at your fingertips, so you can get through problems using "slick" intuitive arguments based on "shortcuts". (Compare to "elegance" in proofs)

Finally, do every solved problem you can get your hands on.

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