# Soft question: How can I make sure that I understand material without any practice problems?

I'm a third year undergraduate student getting ready to do research over the summer in computational fluid dynamics. My background reading is a text by Chorin & Marsden.

This is my first experience with a graduate school level text, and I'm finding the difference to be quite significant. I've felt quite comfortable with all my undergrad books, and really find the examples, illustrations, and practice problems to be helpful. It seems, however, that graduate level texts leave a lot of that sort of thing out. I have seen exactly zero examples in this text so far, and there is virtually nothing helping me gain any physical intuition.

My two questions are:

1. How can I make sure I understand the material without having much in terms of practice problems? In the same vein: I find that seeing an example really helps me understand what's going on. How can I achieve the same level of understanding without examples?

2. Why are graduate level texts so terse? I imagine that part of the answer lies in the fact that higher level material is less focused on computations and more on concepts and proofs. Even still, I can't see how examples and trying to give some physical intuition could possibly hurt.

Any other comments on how to make this process a bit easier are, of course, greatly appreciated.

• Plenty of graduate level texts have examples and exercises. This is not a difference between undergrad vs grad book – user399601 Apr 9 '17 at 17:03
• Have you tried to find the sources cited in the preface? – Michael McGovern Apr 9 '17 at 17:59
• I have graduate textbooks that have dozens of problems at the end of each chapter. I don't have an answer to your question though. – Matt Samuel Apr 9 '17 at 19:20

In addition to what Kyle said, using the Feynman technique is really a great help. That is: If you just read some theorem, can you explain it simply, without using any complicated terms and by using analogies. Try explaining it so simply that you could explain it to a first year undergraduate and they'd be able to understand. Also try to prove theorems yourself. In good texts, you'll often find that you're asked to prove the theorems of the next chapter in the exercises of the preceding one.

• Even more so, if you can find a text that is well written, but leaves a lot of details to the reader, these can be great for learning. For example, B.V. Shabat's Complex Analysis in Several Variables text essentially forces you to write the whole proof out yourself since so much is assumed. – Very Confused Apr 11 '17 at 23:08

A great exercise that is not an 'exercise' is to add all the necessary details that are implicitly assumed or left to the reader by the author. Also, as a mathematician, I believe it is our duty to ask questions and create 'exercises' while we read. For example, suppose you are reading a text on elementary group theory. For example, the author presents the definition of a normal subgroup. That is,

Definition A subgroup $N$ of a group $G$ is a set $N$ closed under conjugation by elements of $G$. More precisely, it consists of all elements $h$ such that $ghg^{-1} \in N$.

Then, without reading further, ask yourself whether you can come up with some examples of normal subgroups. Then ask yourself whether you can determine any facts about normal subgroups, and whether you can actually prove or disprove. Suppose you have done this for a fair while, then you can read a little further and read a theorem. For example,

Theorem The kernel of any group homomorphism $\varphi : G \to H$ is a normal subgroup of $G$.

Then instead of passively reading the proof presented, prove it yourself. Then ask yourself, what conditions can I relax or add, and what changes?

For example, what is the kernel of an isomorphism? Then you ask, did I actually need this isomorphism condition to obtain the result. Actually, all we needed was that the homomorphism was injective. And proceed from there. At the start this can all be quite difficult, but it's worth it in the long run, and you get used to it pretty fast.

Best of luck.