> Most areas of CS you do are unlikely to need advanced calculus or topics that depend on it (differential equations, topology, measure theory...).

Right. The only advanced calculus text I listed was 'Baby Rudin', and the main contribution there is just to get some of the more important properties of the real numbers, Euclidean n-space, infinite sequences and series, and Riemann integration solid.

If a CS student is to stop before these topics, okay, but if they are going to go on then these topics will be part of what is generally assumed.

For CS, as we know, it is doing what EE did -- moving beyond its core tools and into what to do with those tools. E.g., EE got into nonlinear filtering and stochastic integration. E.g., CS is now getting into both optimization and statistics. Then being handy with that Baby Rudin material, at least the early chapters, will get to be important.

With Baby Rudin already done, there can be a good course in differential equations, and such a course can be a good way to see some of the value of what did in linear algebra and Baby Rudin and to exercise that material. At some point in the future, a CS guy might well get into some work involving differential equations -- viral growth models, flight of airplanes and space vehicles, and much of mechanical engineering.

One of the main themes in the future of computer applications is handling 'randomness', and my view is that serious work in that direction should have the measure theory foundations. I did an A/B on that! Early in my career I tried the easy way. After measure theory, Neveu, Breiman, Loeve, Dynkin, Lipster, Shiryaev, etc., I concluded that the measure theory approach to probability, stochastic processes, and statistics was essential.

In particular, without measure theory, people too easily get totally stuck in the mud on what 'random' means, while with measure theory Kolmogorov has a really nice answer. My view is that people may not like Kolmogorov's answer but that, with some really simple assumptions, we get forced into that answer anyway!

For the rest, the Hilbert and Banach spaces won't go away! E.g., a huge dessert buffet of really finger lick'n good applications of the Hahn-Banach theorem is David G. Luenberger, 'Optimization by Vector Space Methods', and that book will be a nice source of methods for a lot that computing people might encounter.

> You left out graph theory and combinatorics, both of which are extremely important to CS.

For combinatorics, I assumed that one would get enough, in some ways deeper than in Knuth's TACP, from abstract algebra and elementary probability. E.g., combinatorics is a lot about counting, and so is group theory in abstract algebra.

For graph theory, I assumed that one would get enough from optimization on networks. E.g., a 'basis' in the network simplex method on a network is a minimum spanning tree! And the max flow/min cut theorem can follow just from linear programming. Dynamic programming, which I mentioned, can be viewed as graph theory.

I did mention linear programming (and so does CLRS), and one reason is that in the CS study of algorithms the algorithms for linear programming are important, and surprising, benchmarks. Also integer linear programming is one of the more important motivations for the question of P versus NP.

For some of the surprise, the simplex algorithm has low degree polynomial expected performance (K. Borgward) but exponential worst case performance (Klee and Minty), but the polynomial algorithms (Khachiyan) when they are faster than simplex have both of them too slow for practice! So, at one of the first places we looked at computational complexity for problems more challenging than, say, heap sort and AVL trees, an exponential algorithm turned out to be superior in nearly every sense of interest to a polynomial algorithm! There never was a guarantee that the study of computational complexity would be easy!

But there's a lot of overlap: I didn't mention courses in 'finite mathematics', combinatorics, or graph theory, but one way and another what I described should provide enough coverage. I didn't mention the CS book CLRS, but I mentioned good coverage of some of the more advanced topics in that book. Or, there is a lot of blending old wine and pouring it into new bottles.

There is a broad point: A major theme in CS now is to borrow, modify, and apply work done some years ago in applied math, especially from operations research, e.g., linear programming, flows on networks, stochastic point processes, and statistics. While CS has some new applications, typically the material is done more carefully in the old applied math sources. So, I emphasized learning the material as math instead of as CS. Besides, the OP was asking about math for CS and not 'mathematical CS'! Also the title asked for the "hard way"!

There's another broad point: What to learn and why to learn it? Just a first cut view of what O( n ln(n) ) means should take only a little searching on the Internet. Besides, Knuth's TACP is quite clear on such 'asymptotics'. I wish Microsoft's MSDN documentation of .NET was as clear and easy to read as TACP!

One reason not to learn this stuff is to confirm that Knuth, Sedgewick, CLRS, etc. were correct after all!

Generally the reason to learn such stuff is for new applications in the future. For that, my view was just to stay with a relatively traditional course in relatively applied math although I leaned away from classic mathematical physics and more to business applications.