How should I learn from proofs in Applied Mathematics?

Question

I am aware that similar questions have been asked here and elsewhere about how to learn from proofs. Some common advice is:

1. Most proofs are written in a polished form, not how they were first discovered. Look at the polished proof and try to figure out how it was first discovered.

2. Don't just try to understand a proof line-by-line. Instead, try to capture the main ideas and retain them, instead of retaining the details.

3. Try to figure out the proof on your own, and use the book proof as a hint.

4. Try removing one hypothesis at a time, and finding counterexamples.

This is all very good advice, and I have used all of it when studying pure math. However, I have recently changed to studying applied math, and I am unable to apply these strategies sucesfully most of the time. I will try to explain why:

Pure math seems a lot more clean. Take the Sylow Theorems, or the Heine-Borel Theorem as an example. Their proofs may be very tricky to come up with from complete scratch; but you can summarize the proofs in 2-3 key steps, and if you remember these, it's not difficult to reproduce the entire proof. These theorems also have relatively few hypotheses, and its not too difficult to come up with counterexamples if you remove certain hypotheses.

The proofs in applied math are very different. First, they often have many more technical hypotheses; "this constand it less than $1/2$, this variable is bounded by this complicated function", etc. Therefore, it very difficult (and to me, unenlightening) to try to come up with counterexamples which show the neccesity of these very specific hypotheses.

Secondly, the proofs often consist of a lot of heavy manipulations that are very difficult to remember. At each step, you may have 2-6 manipulations that you can consider: Taylor expand this to first order, Taylor expand that to second order, use Triangle Inequality here, make this substitution there, etc. If the proof is 4-5 steps, there may have been 20-50 wrong routes that you could take. This makes the proof both very difficult to remember, and very difficult to come up with.

To illustrate my point visually, here is a proof from pure math that I am used to, and here is a typical proof I encounter in applied math:

Example of pure math proof:

Example of applied math proof:

Answer 1

If you were to advance in Pure Mathematics, you would also find that the theorems get more technical, with all kinds of messy hypotheses. What you are seeing in Pure Mathematics are results from a century or so ago. Many books have been written about them, and the has been a lot of time to clean up the results and proofs.

Answer 2

It may also be worth mentioning that in the kind of result you're seeing in "Applied Math" (which, as Stephen remarked, you might also see in "Pure Math"), the statement of the theorem is developed at the same time as the proof.
The process might go something like this.

We want to prove some conclusion, say $\lim\inf_{k \to \infty} \|g_k\| = 0$, under some conditions.
What should those conditions be? It's usually not realistic to expect an "if and only if" condition, but other things being equal it's better for our theorem to be as widely applicable as possible, and maybe we have some examples in mind that we want to be covered. We develop an outline of how we might expect to prove the conclusion for something like our examples, and along the way we see what conditions need to be true in order for this to work. Now look at each of these conditions. Might it follow from something else? If so, deriving that condition will become part of the proof. If not, the condition becomes one of the hypotheses.