Name: Advanced calculus for applications
ISBN: 9780130111890

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graycatAug 2019

Most of the best math is not trivial and, thus, usually takes some effort to understand.

There are some good authors of math for, say, calculus, linear algebra, differential equations, advanced calculus, advanced calculus mostly for applications, real analysis, optimization, probability, some topics in stochastic processes, introductory statistics, various more advanced topics in statistics.

Mostly the books are short on motivation and applications, and as a result it is too easy to spend time on material likely not worth the time unless you can be sure both to live forever and remember forever.

For calculus I liked Johnson and Kiokemeister. I taught from Protter and Morrey, and it was easier than J&K. Lots of people liked Thomas.

For linear algebra, I liked E. Nering and, then, P. Halmos, Finite Dimensional Vector Spaces which really is baby Hilbert space theory. Take Nering seriously -- he was a student of E. Artin at Princeton. His treatment of linear algebra is balanced and polished. For one of his editions, he has some group representation theory in the back, good, and some linear programming, really bad.

A lot of people like the MIT Strang book.

For advanced calculus to help when studying physics, especially electricity and magnetism and engineering, I very much liked

Tom M. Apostol, 'Mathematical Analysis: A Modern Approach to Advanced Calculus', Addison-Wesley, Reading, Massachusetts, 1957.

He has more recent versions, but for physics and engineering I like the 1957 version and don't like the later versions at all.

For ordinary differential equations, I liked

Earl A. Coddington, 'An Introduction to Ordinary Differential Equations', Prentice-Hall, Englewood Cliffs, NJ, 1961.

He makes variation of parameters look really nice -- then can understand the remark in the old movie The Day the Earth Stood Still. Ordinary differential equations is a huge, old field, and there is some question about how much of that deserves study now. Do notice that for systems of ordinary differential equations, get to apply some linear algebra in cute ways.

For advanced calculus for applications, there is the old MIT Hildebrand -- he knows what he is talking about, is easy enough to read, and a good place to go if need one of his topics.

In recent decades, the pure math departments wanted to teach advanced calculus as the theorems and proofs for freshman calculus. So there is Rudin, Principles of Mathematical Analysis, third edition (not the first two, maybe a later edition if there is one). So here's what is going on: He wants to develop the Riemann integral which is the one in freshman calculus. For that he wants to integrate over a closed interval on the real line, that is, some [a,b] which for real numbers a <= b is the set of a real numbers x so that a <= x <= b. Rudin will argue that [a,b] is a compact subset of the reals and show that the Riemann integral exists on all compact sets. So, the first chapters are big on compact sets. Then he talks about functions that are continuous and then ones that are uniformly continuous. With uniform continuity, the Riemann integral follows right away. Later he does some infinite sequences and series and then uses these for careful treatments of some important results, the exponential function, the number e, the sine and cosine, etc. Later he does integration of functions of several variables on manifolds for Stokes theorem and the related divergence theorem, a fully careful treatment of these theorems used in E&M. He does the Cartan exterior algebra: What is going on is that he wants to integrate a function g: M --> R where M is a manifold, that is, the range of some function f from some box, triangle, etc. to the space with the M. So, for this need the formula for change of variable for integrating with several variables, and that is a determinant of a square matrix. This integration is a multidimensional version of the line integral where direction of integration is important -- the exterior algebra is the multi-dimensional version of that. Can see that again in some treatments of general relativity in physics.

I like Rudin's third edition: Once know what the heck he is driving at and how he is getting there, say, as above, then his high precision is welcome.

For statistics, I suggest using some popular elementary book as a start. Then learn probability really well and from then on study particular topics in statistics as needed. The current directions in machine learning promise to make lots of particular topics important.

For a first book on statistics, consider

George W. Snedecor and William G. Cochran, 'Statistical Methods, Sixth Edition', ISBN 0-8138-1560-6, The Iowa State University Press, Ames, Iowa, 1971.

My wife did really well with that. So, get a good start on statistics and, then, get to learn some analysis of variance (experimental design), an underrated topic.

For a second book on statistics, consider

Alexander M. Mood, Franklin A. Graybill, and Duane C. Boas, 'Introduction to the Theory of Statistics, Third Edition', McGraw-Hill, New York, 1974.

Here, go quickly and get only the high points and don't expect the math to be very good -- in places it's pretty bad.

For regression analysis and linear multivariate statistics more generally, there are several books, Maurice M.\ Tatsuoka, Donald F.\ Morrison, William W.\ Cooley and Paul R.\ Lohnes, N.\ R.\ Draper and H.\ Smith. So, in particular, get enough to understand that regression is a perpendicular projection and, thus, get the Pythagorean theorem again.

For more on such statistics aimed at machine learning, get the Breiman CART -- Classification and Regression Trees, maybe much of the start of ML.

With that much in statistics, will have seen a lot of applied probability and may be ready for the real stuff. For that, need measure theory, e.g., the first half of Rudin, Real and Complex Analysis or Royden, Real Analysis. Then read Breiman, Probability. After that might read some of Chung, Loeve, Neveu, and maybe some more. Then return to applications including statistics with a really solid foundation in probability, random variables, the classic limit results, and much more. Then can read and/or write lots of advanced topics in statistics.

For optimization, a similar review is possible, but it's getting late.

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graycatDec 2014

Take the standard path.

High School

Algebra I

Plane geometry (with emphasis on proofs)

Algebra II

Trigonometry

Solid geometry (if can get a course in it -- terrific for intuition and techniques in 3D)

College

Analytic geometry (conic sections)

Calculus I and II

Linear algebra

Linear algebra II, Halmos, Finite Dimensional Vector Spaces (baby version of Hilbert space theory)

Advanced calculus, e.g., baby Rudin, Principles of Mathematical Analysis -- nice treatment of Fourier series, good for signals in electronic engineering. The first chapters are about continuity, uniform continuity, and compactness which are the main tools used to prove the sufficient conditions for the Riemann integral to exist. At the end Rudin shows that the Riemann integral exists if and only if the function is continuous everywhere but on a set of measure zero. But what Rudin does there at the beginning with metric spaces is more general than he needs for the Riemann integral but is important later in more general treatments in analysis. Rudin does sequences and series because they are standard ways to define and work with some of the important special functions, especially the exponential and sine and cosine further on in the book. The material in the back on exterior algebra is for people interested in differential geometry, especially for relativity theory.

Ordinary differential equations, e.g., Coddington, a beautifully written book, Coddington and Levinson is much more advanced) -- now can do basic AC circuit theory like eating ice cream.

Advanced calculus from one or several more traditional books, e.g., the old MIT favorite Hildebrand, Advanced Calculus for Applications, Fleming, Functions of Several Variables, Buck, Advanced Calculus -- can now look at Maxwell's equations and understand at least the math. And can work with the gradient for steepest descent in the maximum likelihood approach to machine learning.

Maybe take a detour into differential geometry so that can see why Rudin, Fleming, etc. do exterior algebra, and why Halmos does multi-linear algebra, and then will have a start on general relativity.

Royden, Real Analysis. So will learn measure theory, crucial for good work in probability and stochastic processes, and get a start on functional analysis (vector spaces where each point is a function -- good way to see how to use some functions to approximate others). Also will learn about linear operators and, thus, get a solid foundation for linear systems in signal processing and more.

Rudin, Real and Complex Analysis, at least the first, real, half. Here will get a good start on the Fourier transform.

Breiman, Probability -- beautifully written, even fun to read. Measure theory based probability. If that is too big a step up in probability, then take a fast pass through some elementary treatment of probability and statistics and then get back to Breiman for the real stuff. Will finally see what the heck a random variable really is and cover the important cases of convergence and the important classic limit theorems. Will understand conditioning, the Radon-Nikodym theorem (von Neumann's proof is in Rudin, R&CA), conditioning, the Markov assumption, and martingales and the astounding martingale convergence theorem and the martingale inequality, the strongest in mathematics. So will see that with random variables, can look for independence, Markov dependence, and covariance dependence, and these forms of dependence, common in practice, can lead to approximation, estimation, etc.

Now will be able to understand EE treatments of second order stationary stochastic processes, digital filtering, power spectral estimation, etc.

Stochastic processes, e.g., Karatzas and Shreve. Brownian Motion and Stochastic Calculus. Now can get started on mathematical finance.

But there are many side trips available in numerical methods, linear programming, Lagrange multipliers (a surprisingly general technique), integer programming (a way to see the importance of P versus NP), mathematical statistics, partial differential equations, mathematical finance, etc.

For some ice cream, Luenberger, Optimization by Vector Space Methods or how to learn to love the Hahn-Banach theorem and use it to become rich, famous, and popular with girls!

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graycatApr 2021

You might be correct in part or in whole. We'd have to discuss back and forth! My (1) and (2) will have to be at best only rough, and to save space I omitted the real examples I have in mind to formulate those two. Writing clearly, briefly, and precisely about big subjects is not always easy!

The time I encountered Strum-Louville theory was a course from the fairly well known Hildebrand, Advanced Calculus for Applications from MIT and by a professor with a recent MIT Ph.D. By the way, that book is available in PDF on the Internet for free. To me the topic looked like the math of vibrating strings, etc., but the course did not go deeply into Sturm-Louville theory or anything else. So, from that course I have regarded Sturm-Louville theory now as in (2); but if there is some continuing, deep work there, maybe it belongs in (1). Maybe I touched on that subject once more; talking to M. Athans at MIT, he explained that the optimal control problem I was considering was a "two point boundary value problem with mixed end conditions". So, maybe there are connections with control theory.

For Erdős, I have just regarded him as all in (1), but he seemed to have gotten problems from anywhere, including from outside math so might be in (2). Sooo, for the problems outside math, I meant relatively practical problems, ones where solutions would a lot of value, financial or scientific, for fields outside of math. I've always regarded Erdős as working a lot on interesting, tricky, puzzle problems. But I don't know Erdős's work at all well.

Maybe I can summarize: Back when I had one foot in each of (1) and (2), my definition of applied math was like a recipe for rabbit stew -- "First catch a rabbit." So, first get an application, i.e., from outside math, where an application solves a problem that in some sense is pressing and practical.

Examples: At FedEx, how to schedule the fleet? How to project revenue for years in advance? How best to climb, cruise, and descend an airplane? How to plan an airplane tour under uncertainty. Back in military work, how to use the Navier-Stokes equations to design ship propellers? How to process the data to do beam forming from arrays of passive sonar? How to evaluate the survivability of the US SSBN submarines? How to target missiles? How best to search for a submarine? In a simulation, how to generate ocean waves with a given power spectral density? How to respond to ill-conditioned matrices in regression problems? How to do least squares spline fitting, including for multidimensional data? For each of these problems, there were people who cared a lot about getting the solutions -- these were applications. Puzzle problems seem to be applications of a different kind.

Maybe now there are some good applications to be made analysis of DNA.

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graycatJan 2020

Yes, to quote, as I wrote,

> notes to get a nephew of 9

I'm writing the notes for my 9 year old niece. So, the notes are for a boy of 9. And it's calculus, for a boy of 9. Of course it's "calculus 1".

I thought I mentioned it was for a boy of 9.

So, I omitted (1) a real valued function of a real variable with a compact domain has a Riemann integral if and only it is continuous on a set of measure zero, (2) for such function, if it is differentiable, then it has no jump discontinuities, (3) there is such a function that is differentiable but whose Riemann integral does not exist.

But, still, it is amazing how easy it is to get 36 decimal digits of e, and 500 if want.

I've been through calculus, advanced calculus, advanced calculus for applications, differential equations, local series solutions to the Navier-Stokes equations, exterior algebra, real analysis, measure theory, and more, have taught calculus in college, applied it in US national security and business, and published in it, and still I'd never seen a clear treatment of how easy it is to get so many digits of e from Taylor series.

The results I found are amazing, and my good and long experience indicates that only a tiny fraction of calculus students appreciate that.

And as we know,

e = lim_{i \rightarrow \infty} (1 + 1/i)^i

and it turns out that that iteration is painfully slow, and from my experience this fact is also amazing and poorly known.

It was amazing stuff and a productive weekend.

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graycatMar 2019

I suggest don't let the K-12 and college educational systems make calculus a mess.

Instead, if have some algebra and trigonometry from high school and want to learn calculus, then just get one or a few good freshman COLLEGE calculus books and work through them -- at each lesson or section, read and think about the material and then work all the more challenging exercises. Check answers in the back of the book, a copy of the Instructor's Guide, on the Internet, etc.

For the books, get mostly old ones known for decades to be good. Get just good, used copies -- the subject hasn't changed much in decades. For the books, DO get ones that are good on (A) the completeness property of the real numbers ("Calculus is the elementary consequences of the completeness property of the real numbers."), (B) limits, (C) the epsilon-delta definition of limits, (D) the epsilon-delta, limit definitions of the derivative and the (Riemann) integral, (E) applications. Get more than one such book, use the one that looks the best as your primary source and use the others for alternate explanations and more exercises.

That's what I did: I got a good book and worked through about half of it. Then for calculus in college, I asked to skip freshman calculus and start on sophomore calculus, the rest of the book. A prof gave me a little oral exam, define the derivative, with some TeX notation

f'(x) = d/dx f(x) = lim_{h --> 0} (f(x + h) - f(x))/h

and I was in. I did well, made As both semesters. Went on to advanced calculus, ordinary differential equations, advanced calculus for applications, real analysis, functional analysis, real applications, peer-reviewed publications, teaching calculus, etc. E.g., for an application I derived and used

y'(t) = k y(t) (b - y(t))

to please the BoD at FedEx, keep a crucial investor from leaving, and save FedEx from going out of business.

From what I've seen of high school materials for calculus, I'd advise trying hard to avoid them -- again, just start with one of the best college texts. The one I used for the actual course was Johnson and Kiokmeister, then also used at Harvard, now ancient but still fine.

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graycatSep 2018

With calculus, you are already well into ordinary differential equations -- partial differential equations are different, but with calculus you have a good start on the basics of those, too.

A simple ordinary differential equation is below where of course just from calculus

y'(t) = d/dt y(t)

and the equation is

y'(t) = k y(t) ( b - y(t))

So, for the context: t is time, say, in seconds. y is some real valued function of t, that is, y(t), b and k are constants. We are given the value of y at 0, that is y(t) at t = 0, that is, y(0). We want the value of y(t) for t > 0.

Okay, with that, just need to use freshman calculus, for positive time s, integrate y'(t) from 0 to s. This is a simple exercise, without looking up the last dozen times I did that, maybe use integration by parts or some such. End up with a quotient with some exponentials.

Differential equations pop up in motion, e.g., from Newton's second law, AC circuit theory, and some other areas of science and engineering.

Boundary value problems, e.g., vibrating stings, parts of deterministic optimal control, are closely related but, still, significantly different.

Long some of the pure mathematicians went, in a word, "nuts" studying differential equations. The best of the results are good, and some of those are nicely useful. In places the work has nice contact with linear algebra, matrix theory, and vector spaces of functions, functional analysis, e.g., Hilbert and Banach spaces. But hanging over the whole subject is a suspicion that, really, as nice as the general theories are, mostly the applications are just a few, standard differential equations. It's a little like learning everything about civil engineering when really are only going to do framing carpentry, hang drywall, and apply roof shingles.

Once I bought

Garrett Birkhoff and Gian-Carlo Rota, {\it Ordinary Differential Equations,\/} Ginn and Company, Boston, 1962.\ \

I looked through it, saw lots of intricate stuff, but wondered just why I should dig into that. Since then I read a story about Rota about how, apparently, he felt much the same about the material, got stuck teaching the differential equations course because he wrote that book, and wanted, essentially, to f'get about that book and its material!

I had a full college course in ordinary differential equations. Okay: It left me wildly over educated for the differential equations in AC circuit theory. Otherwise I didn't much like the book, the teacher, or the course.

On the advanced stuff, here is some more

Earl A.\ Coddington and Norman Levinson, {\it Theory of Ordinary Differential Equations,\/} McGraw-Hill, New York, 1955.\ \

It has a nice result of Caratheodory, but in general could lose a lot of sleep working through that!

I had a course from a Ph.D. from MIT from the book, apparently long a standard at MIT,

Francis B.\ Hildebrand, {\it Advanced Calculus for Applications,\/} Prentice-Hall, Englewood Cliffs, NJ, 1962.\ \

So, yes, can find out about solutions via infinite series and boundary value problems. The book was very short on proofs, and to take such material seriously I wanted to see the proofs. Now that I know a lot more math, no doubt some of it originally motivated by material in that book, maybe I could fill in the proofs.

When I was at FedEx, I wondered about the cheapest way to climb, cruise, and descend the airplanes, had heard about

Michael Athans and Peter L.\ Falb, {\it Optimal Control:\ \ An Introduction to the Theory and Its Applications,\/} McGraw-Hill Book Company, New York, 1966.\ \

and flew up to MIT and met with Athans, got his course notes, etc. He explained that an application would be a "two point boundary value problem with mixed end conditions" -- okay, I'd had a course on numerical methods for that. But, in the early parts of the book will see something interesting -- fast, and well written coverage of the differential equations material needed for the book. This is an example of a general situation: Sometimes the best place to learn something is in an introduction or appendix written by a real expert who is also a good writer, intended as background for the rest of the book. So, such a source cuts out the tangential, maybe curious cruft can't much hope to use.

At one point after college on my own I carefully read, not nearly new at the time (TeX markup):

Earl A.\ Coddington, {\it An Introduction to Ordinary Differential Equations,\/} Prentice-Hall, Englewood Cliffs, NJ, 1961.

Coddington was not just a grand expert in the field but also a good writer. I really liked his stuff on variation of parameters -- a bit amazing. Note: Can find mention of that in the famous movie The Day the Earth Stood Still -- apparently that math was hot stuff in applied math about when the movie was made.

Can say some quite similar things about partial differential equations -- e.g., there are deep books, some connections with functional analysis (and even the theory of distributions) but the main interests are the partial differential equations of mathematical physics, especially, Maxwell's equations, the heat equation, the wave equation, Schrödinger equation, a wave equation, and the notorious Navier-Stokes equations -- which likely should attack only for limited goals and in somewhat special cases.

Net, unless you have some significant reason for more, I suggest you learn what you need to know, just in time, when and if you need it. But, in that case, as elsewhere, a good pure math background in calculus, and advanced calculus with the proofs, etc. will be good to have.

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