Improving Math Skills for Physics Students: Where to Start?

[go_ducks]

Hello, I am a 29 year old trying to transfer to a UC or other college of some type to study physics.

My problem seems to be really bad math skills. I know that some of the issue is self-confidence which can go a long way to conquering a problem. But I've just been through a introductory quantum mechanics class (final in one week from today, snarfff..), and I pretty much got served by the simplest examples we did. That is after taking math 15 (first course in differential equations) and while concurrently enrolled in linear algebra. The homework was too hard - it was full of super-hard looking math- and the exams upped the ante . I just got squished. Yep. That's the word for it - squished. Felt like I got thrown into a rough sea. And its made me feel really thick and stupid.


So anyways. Without trying to sound too depressed (I'm not sad about it, really, I was taking a chemistry class and working so time was a factor) I do believe I have some sort of serious issues with mathematics. I will do mindless mistakes like mess up the minus signs, just working my algebra. I write numbers the wrong way around as well. I was in my linear algebra exam trying to find a matrix that will convert from one basis to another,and I got everything back to front and it pooped out pretty badly. I studied it carefully I thought... I don't know how I ended up doing it wrong in the exam. I did the freaking homework and reviewed it . That's just the tip of the iceberg as far as my math ineptness goes. Any time I see things like differential operator arithmetic (we use operators quite a bit in Q.M.), I just don't understand why that proceeds one bit. You can just move a d/dx around onto whatever you want, multiply both sides by them, etc? (Don't answer that question... . its an example of how confused I feel is all.)



So let me throw some question out there. Let's suppose you are, one of those countless physics students that professors complain "they don't have the background mathematically". Then what would you do about it? Because I can't think of anything. I bought shankars basic math for science students, figured I'll try to work through it, I dunno, I wondered if there is an online test you can take to rate your math skills and help you figure out what level you are really at and what you will benefit the most from studying. I really don't know where to start. I wish being able to move y's and x's around was all I had to worry about =p.

 

[CompuChip]

About your messing up minus signs and such, I can only say: work carefully! Even my professors tend to do that wrong (some of them don't even bother, and make "check the signs" a DIY exercise ).

As for improving your math skills, apart from gaining some self-confidence as you said, the trick is to PRACTICE! Probably you've been told a hundred times before, but it's really true: you don't learn math from a book, you only learn math by doing it yourself. If you don't immediately see some step in a calculation, take a piece of paper and write it out in full detail. Yes, that can be boring but it will help you a lot.

Also you need to be aware that physicists tend to be sloppy with their mathematics. Indeed for a lot of purposes, you can treat d/dx as a fraction which you can multiply, invert, pull apart, etc. However, when using it, it helps to remember what precisely it is you are doing. Instead of "multiplying by d/dx" you can say "taking the derivative w.r.t. x on both sides" - in notation it is the same but the latter makes it clear what you are doing.

In general, I would say, keep asking questions. These forums are really useful, as are your class mates. Usually, your teacher don't mind if you drop in sometime to ask a question either, as long as you show them that you have really given the problem some thought.

 

[maze]

For the minus sign and other simple errors, aside from just "being more careful", a good way to reduce them is to employ different strategies for checking your work. If the answer is supposed to be positive and you have it negative... that's a problem! If you are adding a length to a velocity... that's a problem! If your answer is 10km and you expect it to be around a milimeter... you get the idea. Try plugging in simple values and see if the result you get makes sense. Basically go guerilla style and try to check your work using any tools or intuition you have. The more different the check is from the way you solved it, the better.

For the other specific questions, the best thing you can do is identify exactly what is confusing you, and formulate questions that, if answered, would resolve the issue. Then ask your fellow classmates or post online. Sometimes the simple act of formulating the question will cause you to understand it. It can seem overwhelming when there are so many things confusing you, but keep confident and take it one issue at a time. Its like you are laying siege to a fortress. At first it seems impenetrable, but if you keep the pressure on slowly but surely cracks in the armor will develop and eventually it will yield under your might.

As for your whiny professors ("oh my students don't have the mathematical background. waaaaahh.") don't worry about what they say. To properly understand QM you need functional analysis, which basically no undergrad physics major will have taken before QM (functional analysis carefully formulates the "function spaces" that functions live in, and the "operators" that map between these spaces, such as differential operators). Many (most?) physics professors don't even know functional analysis rigorously. They just wave their hands around enough so that you can solve problems, but not enough to get a deep understanding. That's ok though, because solving problems is useful from a practical standpoint, will help inform your intuition, and will make it easier to understand it rigorously if you decide to pursue the subject at a more advanced level.

 

[dE_logics]

I will do mindless mistakes like mess up the minus signs, just working my algebra. I write numbers the wrong way around as well.

You mean like me.

That's why I even like to DERIVE in a computer...you can edit it, instead of rewriting the whole thing and making it a mess.

I just don't understand why that proceeds one bit. You can just move a d/dx around onto whatever you want, multiply both sides by them, etc? (Don't answer that question... . its an example of how confused I feel is all.)

You need to work on your concepts...while doing that, take maths as physics...all concepts...all theory, then work from theory to the maths stuff.

Lets suppose you are, one of those countless physics students that professors complain "they don't have the background mathematically". Then what would you do about it?

Well...I am one of them, thought I don't get any complaints cause the teaches here are too dumb.

But I realize how bad at math I am.

I'll see an e-book to cover the physics-math.

 

[go_ducks]

Yeah. I find myself in agreement with that. I'd say the math for Q.M. is a bit beyond my level having never taken linear algebra before until now. I mean a couple semesters from now I'm sure I'll get more used to it. That's what the professors promise anyway. Although to me, seems like just beating your head on a block and hoping the block cracks before your skull does.

 

[Klockan3]

Qm needs PDE's which in turn needs linear algebra. Shouldn't those be prerequisites of any QM course?

 

[Cantab Morgan]

Are you bad at Math? Or are you just bad at Arithmetic? They are two very different skills, and one is much more important for Physics.

 

[go_ducks]

I'll agree, but the professors think they can just give you what you need and have you learn it on the fly. They say those courses are "helpful". There are some mathematically-inclined students that can do that, and do it damn well. I just sank . Heh. Think I'll start with reviewing calculus.

 

[Skins]

Here's the catch... Are you understanding the concepts and ideas and just making simple errors, i.e. - instead of + , ba instead of ab, etc. ?

 

[go_ducks]

I'm not sure dude. My experience in calc 3 and linear algebra have been pretty bad. Differential equations was ok. I think it all snowballs though. Like if you are making frequent mistakes over negatives like that, and you are trying to understand a concept with a logic that's a bit elusive (Q.M.'s expectation values requiring dry integration comes to mind), you confuse yourself and end up just feeling stupid. Least that's what I've done. I mean sitting for a few hours and finding out just a negative sign was wrong somewhere, it definitely happens to me. But on the other hand, trying to understand the proof for the associative property of matrix multiplication (AB)C = A(BC) ... I wrapped my head around it eventually.. it wasn't easy. And that's very conceptual. I guess I've just felt a bit wobbly and easily confused since calc 2.

What I'm saying is... I have a math problem and I need some kind of strategy to launch an offense against it. I'll spare the time. Just looking for some help getting started here. Hopefully this book I bought will help .

 

[Tac-Tics]

If you want to be better at anything, math or anything else in life, quit whining about it and try your best to change your situation.

Lots of people think they are bad at math. Like every other subject in school, it's taught completely backwards. Unlike most other subjects in school, the human element is removed. For whatever reason, while English class teaches you about writers and physics teaches you about physicists, mathematics teaches you jack about mathematicians. So most people don't "get" it. They don't know who Euclid or Leibniz was. All you get is a set of symbols, abstract vocabulary, and a bunch of poorly explained calculator tricks.

Even if it were properly taught, math is still a hard subject. It's hard because you have to know all the previous concepts to move onto the next. Imagine that you couldn't even COMPREHEND the role of President Clinton in American history without being intimately familiar with every other US President that preceded him!

Math is heavy on the details. It's easy to get lost and confused in a sea of symbols. It's incredibly easy to make a mistake while doing algebra. Even after years and years of study, you're always going to end up forgetting to copy a term or to flip a minus sign. It happens. Don't get caught up in the details. The human mind can only work with so much. Focus on big concepts.

Quiz time! What is a derivative of a function at a point x?

A) It's \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
B) It's the slope of the line tangent to x on the graph.
C) It's the the relationship between velocity and position.

All three are right. You should ideally know all three, but MOST of the time, you only need to keep C in mind. In the few cases where you need a little more, you can "drop down" into answer B. Very rarely will you ever need the full, technical definition of a derivative. I take that back. You will NEVER need it. It's great to know, but unless you are doing proofs from chapter 1 in your book, you will never need to know it. In fact, if you understand B, you can probably work out A from B logically.

Some big picture concepts.

* Matrices are functions for vectors. They are always linear, too.
* A basis is like the grid on a map of Chicago.
* Derivatives "undo" integrals.
* Integrals in two dimensions are not done over a simple [a, b]-style interval, but are done over any two-dimensional set of points.
* Diff. Eq's is simply algebra where the unknowns are FUNCTIONS instead of NUMBERS.
* Most "real" problems in calculus are not exactly solvable.
* Calculus notation and physics notation is full of ******** that doesn't really make sense. The dx's suck, but you have to live with them.
* Continuous functions are essentially functions you can "approximate" to an arbitrary accuracy. They are the cornerstone of science.

Packaging information like that is much more important than memorizing formulas or even doing the algebra correctly. As long as you maintain a sense of how something relates to the real world, you can check your work and make sure it's not completely off.

If you want to progress in physics, you probably want to get calculus down as solid as possible. Math is full of other cool topics, but learning calculus and D.E. gives you the biggest bang for your buck. If you don't understand your teacher's notation, they are probably abbreviating the problem. Differential operator arithmetic is a good example. If your professor moves around a d/dx and you can't justify why he did it, call him on it after class. (Regardless of what professors say, a student asking non-trivial questions in class is just distracting. You'll get a better answer if you ask privately).

 

[CompuChip]

Another good point brought up by Tac-Tics, which was a bit implicit before, is that math classes don't always help.
As said, you might be solving a differential equation in physics, because the velocity and position are both unknown. Then you don't need to know about the definition of derivative. However, the latter is exactly what a class in (P)DE's will start from. In the end, you will probably have seen a completely rigorous proof of why there exists a solution to your differential equation and to what extend it is unique, but actually solving it will not be much easier than before you took the class.

I'm not saying you shouldn't take any math classes, just don't trust in them too much, like "when I take all the math courses / read all the math books, my problems will be solved".

 

[go_ducks]

Would you guys mind condensing this for me? I'm just looking for a simple "start here" kinda answer to get going on math. Start with calculus.. is that what you are saying?