I am currently studying calculus in Russian and my course book is very rigorous.I used to think that I understand everything but I recently noticed that I only understand the logical steps in proofs of theorems and I actually don't understand all the formulas and theorems intuitively and can't see the motivation in proofs.Is there any way to improve intuitive understanding and really feel how mathematics works ,could you recommend any books?
[Math] Understanding calculus formulas intuitively
calculussoft-question
Best Answer
Look for the real world idea the proof is trying to capture; in calculus real world examples should be fairly reliable in many cases. Once you have this, intuitive hand-wavy reasoning can tell you the direction in which to apply your more formal efforts. For example, the product rule can be seen as follows:
Take a rectangle whose side lengths change through time. Let one side be given by $f(t)$ and the other by $g(t)$. How does the area $A=f(t)g(t)$ change? Extend two perpendicular sides in one direction (to represent changes in $f$ and $g$) to make a slightly larger box, and label the resulting L-shaped area $\Delta A$. This can be split into three rectangles, and we see that
$$\Delta (fg)=f\Delta g+g\Delta f+\Delta f\Delta g$$
Dividing by $\Delta t$ and (this is very handwavy) taking the limit as all the deltas go to zero gives
$$(fg)'=fg'+gf'$$
Is this a proof? Absolutely not. It glosses over a lot, and it only handles positive $f$ and $g$, as well as positive changes. But it gives you a hint of what you should be looking for. You could then take this result of how an intuitive example of product changes and create the formal setup:
$$\lim_{h\to 0}\frac{(f(x+h)-f(x))(g(x+h)-g(x))}h$$
And ask how to get this into something that looks roughly like
$$\lim_{h\to 0} f(x)\frac{g(x+h)-g(x)}h+g(x)\frac{f(x+h)-f(x)}h$$
Which is a natural first guess for the formal form of the sloppy area answer. In fact this doesn't quite work: what ends up working is replacing $f(x)$ with $\frac{f(x+h)+f(x)}2$ and similarly for $g(x)$. But the point is that we know where we want to go, and this makes our lives much easier. We might not get there immediately; how to get there can still be confusing. But this is an amazing help. If you're looking for motivation of proofs, look for the main idea that's being captured by the result of the proof and try to set that up to get an idea of where you're headed without worrying too much about formality. Often times part of a proof will suggest itself, and the suggestion can then be tackled properly. Directionless math is very rarely insightful.