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Harald Bohr is credited with saying that If mathematics does not teach us how to think correctly, it at least teaches us how easy it is to think incorrectly. I couldn't find an "official attribution" but I trust Gert K. Pedersen, who quotes it in the 'Preface to the Second Printing' of his book "Analysis Now" (Springer, GTM 118, Revised printing 1995).


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There are two factors I see a lot that might be at play here. First, it might be that you're having trouble with problem-solving -- that is, you can do all the steps when the instructor is walking through them, but deciding which steps to take on your own is hard. If that's the case, I recommend practicing more with problems; for example, try solving ...


3

I am an engineer, and need to use the "practical" side of these things on a daily basis. Assuming you have a strong grasp of 3D calculus and linear algebra, you probably want to start with a bit of each of the following: Multilinear algebra. This is where, e.g. tensors appear. Key topics are: Multilinear forms (especially bilinear forms, and how to ...


2

I am also learning mathematics but I have few suggestions to you as a fellow learner. Solve as much problems you can. Mathematics is not a spectator sport. You need to get your hands dirty, scribble things on paper etc. Read "How to Solve it" by G. Polya and "What is Mathematics" by Courant . These books will help you in developing some interests which is ...


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I realize that this is a very old question, but I found it from the related questions on this question, and my intuition for Orbit-Stabilizer is not reflected in any of the answers here, so I thought I'd add this answer for future readers. My intuition Orbit-stabilizer is a version of the first isomorphism theorem for $G$-sets. Here's my justification. ...


2

This is much too vague a question and will probably be closed. That said, I will try to help. First, there is no "best way". What works for someone else may not work for you. So you have to experiment with several ways and learn how you learn best. I think that is rather like "machine learning", where the goal is to teach a machine to figure something out ...


2

Since he's good at algebra, you could suggest he try to apply it in geometry when he gets stuck. I don't mean it'll be a feasible method on its own for all geometry; traditional Euclidean methods are often more productive than writing everything in Cartesian coordinates. However, consider your problem example. We want to compute a sum of two grey areas $g_1+...


2

You would have to draw it out. Always draw out the graph. Instead of $X,Z$, think of $X,Y$ (easier to visualize) We have $X+Y > 2.5$ which means $Y>2.5-X$. We also have that the range is $1\leq x \leq 2, 1\leq y \leq 2$ First task is to draw the region bounded by $x,y$, and then draw your probability function and see where they intersect. In this ...


2

Personally, I think understanding why something works in general is much better than learning a solution for a specific question and being to reproduce that answer to a single question. As long as, in the process of copying these answers, you understand why they work, what identities they used, and the like, it'll be a very efficient method. But if you only ...


2

Donald J. Newman’s A Problem Seminar is a classic and a delight, and you will certainly benefit from it. It is not a textbook, because it does not teach advanced theorems - it is specifically intended to get your mind aligned to problem-solving. It has a hundred problems (two or three lines long at the most), then a section with a brief hint as to how to ...


2

Some time ago, I was surprised not to find many untyped & simply-typed lambda calculus interpreters among the answers to this question, so I started working for a while in an educational lambda calculus interpreter called Mikrokosmos (can also be used online). It implements untyped and simply typed lambda calculus (and also illustrates Curry-Howard). ...


1

You are trying to improve your problem solving skills and there's a simple way to do that: Solve more and more problems. When I started learning geometry(similar triangles, more specifically), I remember in most of the problems you had to draw a line parallel to say $AB$, then use the side splitter theorem to find the length of some side, the ratio of $\frac{...


1

I think optimal control tends to focus primarily on control of physical systems (such as guidance of a vehicle) while operations research is used primarily (though not necessarily exclusively) for making "managerial", "operational" or "strategic" decisions (such as where to base the vehicle). Some OR problems (but not all) involve decisions with discrete ...


1

You don't practice enough. You need to take the problem, which you think that you know perfect and try to write a full solution. I am sure that you'll see troubles. Good luck!


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Beside $\bullet$ Polya, How to Solve It Another I can think of is definitely $\bullet$ Tao, Solving Mathematical Problems: A Personal Perspective Though not necessarily related, the $\bullet$ Stein, Mathematics: The Man-Made Universe Is what kindled my interest in mathematics; the approach is rather unique, and the level is definitely appropriate for ...


1

One question I often ask myself in these kinds of circumstances is "how would I have thought of doing that?" or something equivalent. Another is "how did I miss that possibility?" What I am trying to do with these questions is to uncover any mental blocks which are getting in the way of seeing the mathematical possibilities. Alongside technique and ...


1

and you what to find P(X+Z>2.5). How do you determine the integral values you need when you compute this? You need to integrate f(x,z) over the intersection of the support and the criteria: $$\{(x,z): 1{\leqslant}x{\leqslant}2, 1{\leqslant}z{\leqslant}2, x{+}z{>}2.5\}\\[2ex]\text{also writen as}\\[1ex]\{(x,z): 1{\leqslant}x{\leqslant}2, \max(1,2.5{-}x){&...


1

This is definitely a question many students face, and has been discussed a bit on this site as well (see this excellent answer). In general, I think it's unwise to try and study and completely understand/come up with a solution to each and every exercise in a book. You should definitely read the solution after struggling for a while (~$1$h) and (re)reading ...


1

I have taken course on all of these subjects, and here is what I would recommend (since you are asking). First combinatorics, then graph theory, then number theory. Taking number theory at the same time as abstract/modern algebra is a very good idea, since there are lots of connections between group theory and number theory. My justification for this ...


1

I think of an expression as a game whose pieces are the symbols I'm manipulating. The rules of associativity, commutativity, etc let me know what the legal moves are. So when I find I'm being sloppy, I rewrite the entire expression in rows, making only one move between rows. In your example, it might look like this: $$ \begin{align*} & (7a^5b^3)\cdot(5a^...


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I enjoyed Introduction to Analysis by Maxwell Rosenlicht. I consider it a beautiful and elegant work. Some of the problems are rather difficult; but analysis is a difficult subject. I had the pleasure of taking Differential Topology with him as an undergraduate at Berkeley. I thought he was pretty impressive. Also entertaining, with his "I'm getting ...


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At this point, I can just as well make my comment into an answer, and also elaborate a little bit on it, since I think this is good advice for any high school student. Note: I am not familiar with what mathematics they teach at high school in different countries, so this suggestion might be too simple, but at least it shouldn't be too hard, trust me. As I ...


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I read a doctoral thesis on teaching geometry to non-seeing pupils. The most important thing for these children was to touch and compare objects. As for graphs - you can learn without them, however math are much more beautiful and easier with graphs, courbes, figures etc. You are not excluded from the possibility to use them! It is possible to buy plastic ...


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In my experience, to really remember math for a long time, two things need to happen. 1) You need to really understand WHY a mathematical statement is true, in as many different ways as possible. Can you think of a visual explanation? Can you think of another method to solve the problem? Your goal should be to interact with the material in as many ...


1

Sometimes, particularly in maths, feeling that you are a slow learner can be a good thing. Some people demand of themselves that they attain a much deeper degree of mastery of a subject, than others do. While another person may be satisfied that they can apply the methods taught to them, others are only satisfied when they can apply the methods as taught, ...


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