3000 Solved Problems In Calculus

[Narkis Eatman]

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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.

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 Amazon.com where you can preview the book.

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!)

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.

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.

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)

I recommend OpenStax open-resource digital books from Rice University for all levels of college math, from prealgebra through calculus and statistics. For the student, they're free, can be shared and copied without restriction, carried on mobile devices, etc. From the instructor's perspective, one can rely on immediate access by all students, show the content on overhead projectors, remix the content into presentations and handouts, etc. In my opinion they're at the same level as any professional print text, after several years of development, and boast extensive lists of professorial authors, editors, and reviewers (e.g. Calculus primary author Gilbert Strang is from MIT).

Don't be put off by the age. It is the first modern text. Almost all the problems have answers in back. It was actually designed to have double the problems needed (so that instructors could assign a "full" suite for at home work or a full suite for in class).

B. Also there are some popular drill books (10-25 bucks for bound copy) available at Amazon or book stores. I would recommend a hard copy as I hear bad things about digitized math books for legibility.

You may recognize the cover from the movie Stand and Deliver (I used it in HS also). Escalante said nice things about its problems as well in a research article. Note that it has ALL the answers in the back, ideal for the self-learner.

Calculus Problems by Baronti, De Mari, van der Putten & Venturi might be worth a look. It has problems on all the topics you mention, along with basic ordinary differential equations. There is also a final chapter containing problems whose solution may require techniques from throughout the book.

Viazovska solved the sphere-packing problem in 8 and 24 dimensions, which asks about the most efficient way to arrange solid spheres. She is a professor at the cole Polytechnique Fdrale in Lausanne, Switzerland.

Artur Avila (born 1979) is a Brazilian mathematician, and the first Latin-American to receive the Fields medal. He made numerous discoveries related to chaos theory and dynamical systems.

Zhang discovered that there is a number k less than 70 million, so that there are infinitely many pairs of prime numbers that are exactly k apart. This was a groundbreaking discovery in number theory, for which he received the MacArthur award in 2014.

Wiles had been fascinated by the problem since the age of 10, and spent seven years working on it in solitude. He announced his solution in 1993, although a small gap in his argument took two more years to fix.

Adi Shamir (born 1952) is an Israeli mathematician and cryptographer. Together with Ron Rivest and Len Adleman, he invented the RSA algorithm, which uses the difficulty of factoring prime numbers to encode secret messages.

Easley wrote the software for the Centaur rocket stage, and her work paved the way for later rocket and satellite launches. She also analysed battery life, energy conversion, and alternative power technologies like solar and wind.

Sir Roger Penrose (born 1931) is a British mathematician and physicist who is known for his groundbreaking work in general relativity and cosmology. He also discovered Penrose Tilings: self-similar, non-periodic tessellations using only two different tiles.

In his 30s, Nash was diagnosed with paranoid schizophrenia, but he managed to recover and return to his academic work. He is the only person to receive both the Nobel Prize for economics and the Abel Prize, one of the highest awards in mathematics.

Her extraordinary ability to calculate orbital trajectories, launch windows and emergency return paths was widely known. Even after the arrival of computers, astronaut John Glenn asked her to personally re-check the electronic results.

During his life, Erdős published around 1,500 papers and collaborated with more than 500 other mathematicians. In fact, he spent most of his life living out of a suitcase, travelling to seminars, and visiting colleagues!

He was one of the founders of the Bourbaki group, a group of mathematicians working under the collective pseudonym Nicolas Bourbaki. The goal of the Bourbaki group was to unify all of mathematics with a formal, axiomatic foundation.

Weil believed that many problems in algebra and number theory had analogous versions in algebraic geometry and topology. These are known as Weil conjectures, and became the basis for both disciplines. They also have applications in fields like cryptography and computer science.

At the age of 25, just after finishing his doctorate in Vienna, he published his two incompleteness theorems. These state that any (consistent and sufficiently powerful) mathematical system contains certain statements that are true but cannot be proven. In other words, mathematics contains certain problems that are impossible to solve.

During World War II, Kolmogorov used statistics to predict the distribution of bombings in Moscow. He also played an active role in reforming the education system in the Soviet Union, and developing a pedagogy for gifted children.

After a few failed attempts to contact other mathematicians, he wrote a letter to the famous G.H. Hardy. Hardy immediately recognised Ramanujan's genius, and arranged for him to travel to Cambridge in England. Together, they made numerous discoveries in number theory, analysis, and infinite series.

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