# Applied Math Foundational Books

I have a BA in mathematics from a pretty good school, where I effectively exhausted the mathematics sequence. The sequence mostly focused on pure math (including measure theoretic real analysis and some functional analysis). I am now interested in learning about financial mathematics, at an intermediate to advanced level.

I have studied probability to what I would call an "intermediate" level (using Feller, Ross's "Introduction to Probability Models", Gut's "An Intermediate Course in Probability", and I have begun learning probability in a measure-theoretic context. I probably have some gaps in my knowledge here, for example, with branching, recurring processes, and queues, and ....

I am, however, concerned that this is not enough of a foundation in applied mathematics. For example, I have had very little exposure to differential equations (and I couldn't solve one to save my life). I have seen references to stochastic differential equations, so obviously, I should learn about the deterministic kind first.

Can you suggest a list of applied mathematics books that would serve as a foundation for an advanced course in applied mathematics? What I mean is, I suspect I will have to fill in the gaps in my knowledge. What books would you suggest as references/introductions to the field?

Well, if you're serious about applied mathematics-and serious in that you don't just want "reciepe" books,rather applications that build on the meaty theory background you have-then you should avoid such texts and try and locate books that don't avoid theory,but merely downplay it. Those are the "real" applied mathematics textbooks.

You definitely need to strengthen your background in differential equations, you're pretty much dead in the water without that in your background. The best beginning textbook I know on differential equations is George Simmon's Differential Equations with Applications and Historical Notes. It is one of the most beautiful, richest textbooks you've ever find on any subject-it covers all the basics of ordinary and partial differential equations using only basic calculus and a context of not only physical applications,but incredibly detailed and scholarly notes on the historical founders of the subject. It's a must have for any mathematical library. The one flaw the book has is that it doesn't explicitly use linear algebra,but rather old fashioned linear equation notation. But by all means,don't let that chase you away from one of the best books there is on any subject. It really is the book to start with.

Another subject you'll need to be very comfortable with in terms of applications is linear algebra and there's no better book from an applied standpoint for this then Gilbert Strang's classic Linear Algebra With Applications. Be careful with this because there are several versions of this book and the one you want is this one-the other one is much gentler and less substantial. This is the one with the real red meat by one of the great applied mathematicians of our era-with a host of applications you won't find anywhere else,both standard and exotic. While you're at it, check out Strang's calculus text online at his website. It may be the best basic calculus text there is bar none-with more applications then just about any other text in it's wieght class. And best of all, it's free!

For financial mathematics,you really need a good background in probability theory. It sounds like you have a good working knowledge of the brute theory, but that'll only take you so far. You need to back up and learn some of the basics first, particularly the applications. One of the best books that currently exists on this subject for the beginner, which contains a host of applications, is Grinstead and Snell's Introduction to Probability. It provides not only a comprehensive undergraduate course in the subject,but it provides many applications, particularly to finance and the social sciences. Best of all,it's available online for free for download.

That should get you started. I'm sure the others will recommend other good texts as well and if I think of any others,I'll edit this post and add them. Good luck!

• I'm pretty sure my analysis professor used Davidson and Donsig as a supplement to baby Rudin. I've been using Shreve's Stochastic Calculus for Financial Mathematics and Neftci's Introduction to the Mathematics of Financial Derivatives, and have enjoyed them. I also did a brief survey with Aggoun's "Measure theory and Filters" and Bobrowski's "Functional Analysis for Probability and Stochastic Processes". Those were a little too theoretical/tangential for my current goals, but they did have good definitions and are on my bookshelf for later digestion. Jan 15, 2015 at 23:56

I recommend Gilbert Strang's books: 1) Introduction to Applied Math; 2) Computational Science and Engineering.

• +1 I actually would study those books AFTER the ones I recommend in my post,little. But important recommendations nevertheless. Aug 18, 2014 at 1:19

There is an (ongoing) series “Foundations of Applied Mathematics” by Humpherys, Evans and Jarvis. https://foundations-of-applied-mathematics.github.io/

• This is tailored towards data science and industrial mathematics (where most students are likely to end up in the current educational landscape). Nov 1 at 17:14

I recently completed my PhD in applied mathematics in July 2022 and was able to draw a lot of useful information from these textbooks while studying in graduate school as well as outside of it:

1. Applied Mathematics, PDEs, ODEs:

2. Statistics:

3. Computational Mathematics:

4. Fluid Mechanics:

5. Data Science, AI, and Machine Learning:

In general, I enjoy reading textbooks written by Lloyd N. Trefethen, Gilbert Strang, Lawrence C. Evans, Randall J. LeVeque, and Sheldon Axler. The popular YouTube channel The Math Sorceror also has a good overview on math books.