I'm still more or less at the beginning of my journey to become a mathematician, but I already recognized that there is a huge gap relating to my skills between solving problems in linear algebra and solving problems in calculus. I really like calculus, but solving problems in linear algebra seems way more easy to me then solving problems in calculus, and that bothers me - a lot. Most of the problems in calculus look like this:
"Show the identity of ..."
"Show that the limit of ... is ..."
"Solve ..."
"Show inductive that ..."
All of these tasks have something in common: They include a lot of playing with equations. Linear algebra, on the other hand, requires a (how to say it in English?) "structural approach" or "structural thinking". Its more about the relations between different objects then about writing down $15$ different steps until you showed that the left side is indeed the same as the right side of an equation.
But since it would be way too easy to simply accept the fact that I'm not that good at calculus, I want to get better at it. But where do I have to start? Does anyone have useful hints for me?
Edit:
Since this was asked in the comments, I'll give an example of a task I would have my problems with. In this case, it's simply about showing that the limit of the $p$-norm is equal to the right side.
$$\displaystyle\lim_{p\to\infty}\lVert \textbf{x}\rVert_{p}=\lVert \textbf{x}\rVert_{\infty}$$
$\endgroup$ 114 Answers
$\begingroup$I have a few general ideas that might help you develop your skills and improve:
Study the structure of proofs carefully (especially well-written ones!) This will help you understand how to write a proper proof. Pay attention to the logic of the proof, and how each declaration follows from previously established results (whether they be theorems or things demonstrated already in the course of the proof).
The key concepts in analysis are usually buried in the proofs, so try to see past the equations and on to the ideas that are being represented (for example, the derivative of a function at a point is merely the slope of the tangent line of said function at said point). Pictures are extremely helpful here, but take care that the pictures you make actually represent the ideas being discussed.
Topology is very worth learning as it is the modern foundation for analysis as a whole. Several concepts are much easier to understand from a topological viewpoint than from an analysis perspective. (For me, I found the topological definition of continuity to be both simple and easy to understand and apply, to the point that I promptly decided to forget the usual definition seen in Calculus classes because it was unnecessarily complicated and entirely replaceable by the topological definition, another example of topology helping analysis understanding is compactness. It's hard to really appreciate the reason for it's bizarre looking definition until you actually do proofs that use compactness, then you start to understand why the property is actually important.)
My last bit of advice is rather specific: Practice doing arguments/proofs with inequalities and seek to develop and intuition for it. It's immensely helpful with certain things you do often in analysis (like proving limits and converge), so much so the my Advanced Calculus professor lamented that the curriculum leading to Advanced Calculus doesn't spend more time on it so that we would be better prepared (this after many people in the class complained that they had no intuition for how to manipulate the inequalities in our problems in order to get something we could actually use to prove what we needed to prove).
The best suggestion to make is: use us!
Calculus requires more "mechanism" than linear algebra. Even $(x+h)^n-x^n$ is already quite mechanical in that sense. A lot of the business of working your way through calculus is acquiring an instinct of which bits of mechanism to keep, and which to discard. You will, at the beginning, end up with 12-line proofs that could have been half the length; and equally with 4-line answers that end up proving that $0=0$.
It does get better.
To help the process along, I suggest that you pick something specific that you aren't happy with. You may feel you have had to guess more than you should; or that you have taken a circuitous route to a result that should have been easier. Then write it up properly as a question on here. You probably know by now that "Do my homework!" and "Check this proof!" questions are not welcome, but if you point out "here I had to guess, and I shouldn't have had to", or "it doesn't seem right that all these manipulations are required, are they really?", then people reading your question will get inspired, and want to help.
$\endgroup$ 4 $\begingroup$As a follow up to the comments in my previous answer, here is my approach to proving the proposition given as an example by the OP.
I started by taking a look at the definitions of $p$-norm and infinity norm, the images of the unit circles on the wikipedia page for them clued me in to an intuitive picture of why the proposition was true (as $p$ got large, the unit circle started looking more and more like the square that you get with the infinity norm).
Noting that the infinity norm was the max of the magnitudes of the components of $x$, I realized what I needed to show was that the $p$-norm would, for sufficiently large $p$ be dominated by the largest component (specifically, the proportion of $\frac{|x_j|^p}{|x_i|^p}$ for the relevant $i$ and each other $j\in \{1,...,n\}$ would tend to $0$). Which is true because $|x|^p$ is strictly monotone for $x\neq 0$ (strictly decreasing/increasing).
As a result, in the limit, the sum approaches the $(\operatorname{max}\{|x_i|:i\in\{ 1,...,n\}\}^p)^{1/p}$, with the exponent becoming 1 and thus the result is that the $p$-norm indeed converges to the infinity norm (I've swept some details under the rug here, but I think it's pretty clear how this proof works, and that is it actually correct). Notice how I was able to develop that proof off of intuition in a pretty straightforward manner, only using a few things I knew.
$\endgroup$ 1 $\begingroup$I had to do a crash course in Calculus. I was doing some stuff and honestly, it was the trig that was messing me up the whole time. I don't know $e$ and I don't pretend to. But what I did instead was sort of miraculous.$$y = | \text{baseline} - \text{Y-axis} |$$$$t = \text{length of a to b} $$$$x=2$$$$m = \frac{sec(x)}{2}$$$$n = | y - t |$$$$P = (m*2) + (n*2)$$$$L_n = \frac{P}{2} + \frac{n}{2}$$
If you add the $L_n$ answers together, you get an integrand of the entire curve.
If you average them using the $V = L_0 + L_1 + L_2 + ...$ and dividing by the largest integer, $G = n-1$ then you'll have the integral of the equation: $VG$
Finally, if you the differential of a Gaussian equation, you do the first algorithm I made, and replace $x$ with the answer to it. The derivative will be divided into the first answer. This is the differential. And obviously you can keep going in derivatives.
