Need to understand math better - advice?

[ShadowKnight]

I am and have for a very long time been very interested in physics, simply because I have a need to know how things work. I did not however enter into any field relating to physics as a career, but am still trying to understand as much as I can. Over the years I've studied many topics (Cosmology and QM fascinate me the most) and while I get the concepts and to some degree an understanding, I always hit the math roadblock. I didn't take much math in school (wasn't required) and now even my algebra is rusty. I DO however pick things up very quickly in the right structured environment. Which leads to my question:

I would like to gain deeper insight into physics, which means I need to get over that math wall. I am not looking to gain an advanced degree in this field. I'm not trying to become a physicist. I simply need some guidance on what math classes I should take (and what order?) based on the fact that its for my own understanding. Since I'm not going for any advanced degree here I will likely use community colleges, starting with an algebra brush up course. Any suggestions? Thanks.

 

[whozum]

Some algebra courses wouldn't hurt, but calculus definitely, up to vector calculus (3 semesters at ASU). Some Differential Equations would be a good addition, but it depends on how indepth you want to go.

 

[ShadowKnight]

...it depends on how indepth you want to go.

I'm not looking to discover the Theory of Everything, but I'd like to be able to understand it when someone else finally does.

 

[whozum]

Well, how in depth do you want to know about the ToE?

For example, you can understand what relativity is about and its implications without knowing its derivation and the math behind it. The first would require some trigonometry and algebra, the latter would require a lot of calculus.

 

[ShadowKnight]

I probably won't want to focus on the derivation so much as implications, just what is needed to gain a deeper understanding. For example, entropy is something that interests me. I knowwhat it is, I've read many descriptions. I even know the formula for it, S=K log W. But even with what I think is a good grip on entropy, my understanding is limited by not knowing what that formula means. Lorentz contractions are another good example, I know what we use them to figure out but without understanding the math involved, I don't have a deep enough understanding. So what happens is when reading anything, even the "pop-sci" books on physics, I just ignore the formulas in the pages or the appendices, and because I have to do this I KNOW I'm missing out on something.

The tools to figure these things out are right in front of me, I just have to learn to use them. The best way for me to learn to use them is knowing which courses to take and in what order.

 

[whozum]

Perhaps the best thing to do is to ask an advisor at your college to see the best way to go about what you want to learn.

Not understanding S = K log W isn't a problem of math, its a problem of 'how'. Its apparently a model for something, I'm guessing the state of entropy over time. ( I don't know much/anything about entropy ). I'm taking thermodynamics next semester and I'll learn more about entropy than I probably will ever want to learn.

Ask your college counselors, and more importantly do some of your own research online

wikipedia.org is an excellent source of information on scientific concepts, I learned a lot of (and got interested and started studying) physics from that website alone.

 

[ShadowKnight]

...Ask your college counselors, and more importantly do some of your own research online.

Good advice if I wasn't a 30 year old developer who has been out of college for years :) I did go to 2 of the community colleges here and pose this question to them. One of the advisors asked me what QM was and how math would help me understand it. That's why I posted here. Most of my research over the last 10 years has been online (Wikipedia IS excellent) but the problem with trying to develop advanced math skills online is that I won't have anyone right there to ask questions or giude me, and in my case I've learned that I need to learn things like this ina structured manner, otherwise I wander down tangents.

I will be taking that Algebra brush up online, since I'm sure it wil come back to me. But after that I'm not sure what to take.

 

[mruncleramos]

"I'm not looking to discover the Theory of Everything, but I'd like to be able to understand it when someone else finally does." Well if this is the case, then I am afraid most community college classes will not come close to the level of mathematics that is involved with the various theories. (Usually their highest class if differential equations or linear algebra?) And what is with this online learning stuff? Whatever happened to going to university libraries and learning things the old-fashioned (correct) way.

P. S. I really hope that the algebra you are brushing up on is abstract algebra or it will be a while.

 

[ShadowKnight]

I took algebra up through the first year college level, I just haven't USED it in years. My public school system was so bad that we didn't have to take geometry ot trig or anything like that, so I didn't because at the time I had no interest. I can take algebra online simply because I just need a reminder. Anything above that I am going to an actual school for. The only reason I'm not applying to someplace like Stanford or Berkely is because I'm not looking for the advanced degree, so my thinking is a community college is good enough. If that is not true than someone please offer some suggestions.

 

[ktpr2]

Simple. Pick a collegethat looks good. Search their website and see what books they have their students purchase for algebra, trig, calc 1,2,3. By the first book and work your way up. If there's a solution manual for the book buy that as well.

You can hire a tutor from a nearby university for like 10 bucks an hour if you need help.

 

[mathwonk]

normal sequence:

basic algebra (high school, college, i.e. polynomials, exponents).
[book: Algebra, Harold Jacobson]

a little trig,

calculus (this is really the right place to learn trig)
[book: Courant, Differential and Integral calculus, vols I and II]

linear algebra [book: try Shifrin and Adams]

several variable calculus [vol II Courant, or ideally Spivak, Calculus on manifolds, or Fleming: Calculus of several variables, or Crowell, Williamson & Trotter]

ordinary differential equations {Boyce and de Prima?]

THOSE ARE THE BASICS.

After that, for high faluting physics,

abstract algebra (group theory) [try Artin, Algebra]

differential manifolds ( calculus on curved spaces) [begin with Spivak last chapters of Calculus on manifolds]

partial differential equations [ i do not know this subject myself]

operator theory (infinite dimensional calculus and linear algebra)
[perhaps Lang, real Analysis]

YOU CAN GO ON and ON...

complex analysis (calculus over the complex numbers)
[I like Cartan, Complex analysis of one and several variables]

Riemann surfaces (curved one dimensional complex calculus)
[try Siegel, Topics in complex analysis, or Miranda: Algebraic Curves and Riemann surfaces]

complex manifolds (curved higher dimensional complex calculus)
[perhaps Kodaira, or Wells]

algebraic topology, differential topology (groups made of loops and cycles)
[I recommend Artin and Braun]

algebraic geometry, sheaf cohomology, abelian varieties,
[I recommend Shafarevich, Basic algebraic geometry, then Kempf: Algebraic Varieties]

,,, , but let's get real here, at $10 an hour you have spent a lot already, maybe a lifetime... [I am a 60 something math professor and researcher in algebraic geometry, but very ignorant of physics. My idea of learning quantum theory is to read the book by Louis de Broglie my father gave me as a teenager.]

 

[jma2001]

Hi ShadowKnight,

Your story sounds very similar to mine. I too am a thirtysomething IT worker with an interest in physics. After many years of reading popular books, watching science documentaries on TV, and surfing science web sites, I have finally come to grips with the fact that I need to learn the math if I want to truly understand anything. However, I am too old to go back to school, so I am trying to learn as much of it as I can on my own.

Yes, you definitely need to learn calculus. That is a basic prerequisite to just about everything else. Beyond that, one book that I would recommend is "Basic Training in Mathematics" by R. Shankar. It covers many topics in mathematics that are essential to upper-level physics courses, such as multivariable calculus, functions of a complex variable, vector calculus, matrices, linear algebra, and differential equations. Of course, you will not be able to learn all of these subjects from just one book, but it will at least give you an overview of what you need to learn.

As for myself, I am keeping a blog to record my progress in learning physics, and I have written more details there about which books I am using and how they are working out for me. I will refrain from posting a link here, but if you are curious you can send me a PM.

 

[Maxwell]

First off, definitely brush up on algebra, geometry, and trig.

Once you are very, very confident with those topics, move on to Calculus.

Take *Calculus 1, *Calculus 2, *Calculus 3 (Multivariable Calc), *Differential equations, *Linear Algebra, Partial differential equations and a class in *probability and statistics.

While taking these classes you can also start to take early physics classes. Sorry, there really are no shortcuts to learning advanced physics (like QM) without a very strong background in math and early physics classes.

Physics classes you should look into taking:

*Calc Based Physics 1: Mechanics
*Calc Based Physics 2: E&M, Optics, Waves, etc
*E&M Theory
Analytical Mechanics
Modern Physics
Statistical Thermodynamics
Quantum Mechanics 1 & 2

This will give you a very good background in physics. To do more advanced work, like ToE material, you will definitely need to take graduate classes.

I know it sounds depressing, and a lot of work, but you can do it if you are truly interested in the subject. The classes with an * before them are classes you should be able to find at your local community college.

Good luck!

 

[whozum]

You basically justtold him to get a physics degree :)

 

[neurocomp2003]

are you looking to prove some theories or just understand where certain things come from? If your looking to prove some theories...the first book you need in math is aproblem solvig book, its too dark now other wise i'd find my book and tell you the info.

also try wolframs good old mathworld website (i think its www.mathworld.com, but just search wolfram and mathworld...anything you need in math you can find it there with pretty good links to all the basic bkgd you need for a topic)

any 1st year books will do to get a basic understanding of stuff...calculus by stewart is ver good as well as Physics(with modern physics-relativity) by Serway. If you can't successful understand these books(try getting a tutor) than there is no point in moving onto more advanced topics. Also a good book in Differential Equations is necessary, inorder to see some of the equations that they use in physics. I wish i bought the one i used in university.

Now comes the questoin of what particular field of if Astrophysics, I enjoyed my textbook Carroll and Ostlie. But youll also need a relativity book to understand Tensors which is probably the hardest part in astrophysics. Once you get over that hump the rest of that stuff will look like cheese. Some analytical mechanics books have better descriptions of tensors then relativity books

If your into QM...its hard to say...im not a firm believer in QM so i wouldnèt know much about the math...but my textbook was decent. You will need Complex Variables

Particle physics and nuclear physics the math aint very hard...but once you want to get into string theory you might need manifolds and stuff.

you say your a developer...you should try to programme youngs slit model...ive been trying to figure out how to do it myself. This willg I've you a firm foundation on understanding light.

 

[mathwonk]

The only thing hard about tensors is the refusal of most expositions to point out what they really are , and to focus entirely on the notation used for them.

At risk of my sanity I will try again to explain what they are, in a simplified case, that will no doubt evoke objections, but so what.

The simplest tensor is the derivative of a function. Let's take a real valued function f in 2 variables. At each point p it has a linear approximation df(p) which is a linear function of 2 variables, namely df/dx(p) [x-x(p)] + df/dy(p) [y-y(p)]. The coordinates of this function are the two coefficients (df/dx(p), df/dy(p)).

So notationally we have assigned to each point p, a pair of numbers, or a vector
(df/dx(p), df/dy(p)). The fact that this object is linear is used to describe it by some people as a tensor of rank one, or even a tensor of "type" (0,1).


Note that if you expand your funcrtion f about p in a TAYLOR series, then this tensor of rank one is the linear part of the Taylor series.

Now what do you think we should do to approximate f better, say by the quadratic part of its Taylor series? Then we should assign to each point the second order polynomial in the Taylor series. This would assign to each point p a second degree polynomial with coefficients for x^2, xy, yx, and y^2. Of course polynomials are commutative, so we can take the xy and yx coefficients to be equal. I.e. we assign to each point an object with 4 entries which we can abbreviate as a(i,j) for i,j = 1,2, and where in this case a(1,2) = a(2,1).

This assignment of a quadratic polynomial, i.e. a symmetric bilinear function at each point, is called a symmetric "2-tensor". Some people like the symbols better than the real object, hence they refer to the notation a(i,j) as the tensor.

Similarly for every degree, the kth degree homogeneous taylor polynomial of the function defines a rank k symmetric tensor, whose notational representative is a coefficient of form a(i1,...ik), which in our case is symmetric.


Again some people refer to these symbols a(i1,...ik), as the tensor itself.

Now when we change coordinates from x,y to u,v say, then one has a change of variables transformation, given by linear formulas in the derivatives of the u's and v's wrt the x's and y's and vice versa, that can be used to express the coefficients of the u,v terms using the coefficients of the x,y terms. there is nothing more to this than substituting a linear function into each variable of a homogeneous polynomial

i.e. these coordinate transformation rules can be worked out just from knowing the nature of the given tensor as a polynomial, but some people again take these formal transformation rules as a definition of a tensor.


So instead of saying that a symmetric kth order tensor is an assignment to each point, of a kth degree polynomial on the tangent space, they say a kth order tensor is a family of coeffifcients functions a(i1,...ik) which transform by a certain totally unmotivated rule, under change of coordinates.


In my opinion this formal discussion of tensors is responsible for their reputation for difficulty.

now the subject does get more complicated, but only in two ways:

1) we allow also non commutatibve polynomials, i.e. objects whose representing symbols a(i1,...ik) are not equal when we permute the indices. After all, some physical phenomena are non commutative, like rotations in space.
2) we also look at tensors which are kth order bilinear combinations of tangent vectors, and not kth order bilinear functions on tangent vectors, i.e. we look at objects whose transformation rules require substituting the transpose of the derivative matrix, instead of the derivative matrix itself. This let's us represent geometric objects like parallelograms by tensors, as well as tenagent directions.


Now notationally this is a very compllicated business, but that argues to me for de - emphasizing the notational aspect, instead of elevating it to the height of calling it the whole subject.

My point is to keep your head about you when learning, if you want it to occur efficiently. What I have just told you might not be learned in a year of studying some books on tensors. So try to get someone to tell you what is going on, what the the tool is good for, in each case before plunging in. Since you want physics, try to get a physicist to guide you, who knows where you want to get to, and ideally is already there herself.

good luck.

 

[neurocomp2003]

heh i hope you didn't write that up just for me...cuz i know what tensors are after 3 years when i first found out about tensors,its all about finding the right book =]

thanks for typingn it all out though

 

[Bladibla]

You should get yourself a good math textbook. First, a explaining book, and second a theory book.

 

[mathwonk]

well neurocomp, the reason I typed all that out is your comment that learning tensors is the hardest part of a physicist's mathematical education. So I tried to show how familiar they really are, when looked at conceptually. So you were not the intended beneficiary, but rather someone like the original poster.

Indeed you might like to share the titles of some of the books you found useful in learning tensors.

 

[FrogPad]

I'm currently only taking introductory physics courses, (mechanical and electomagnetism) and they are blowing me out of the water. I have a decent math background at this point, but math is something I've strugled with in the past. If you really want to take the easy route you could skip a lot of the calculus courses. If you UNDERSTAND (like really really understand) what integration, you could skip over the process of actually integrating. Maple, mathematica, or a TI-89 will handle the "algorithm" of actually integrating something. You MUST know what it is, or when you get into vector fields it won't make any sense.

Anyways, the same could be said for derivatives. Understand what they are and let a computer handle the algorithm. You HAVE to know algebra, and you HAVE to be pretty damn good at it to. If you're not, it's going to take you awhile. I hated taking algebra courses when I was taking basic algebra. I thought it was hard, and it was just boring... I mean who gives a **** about finding the zero's of a function... long story short, I did not do well in algebra. However, now that I've taken a lot of university math, I actually enjoy algebra. So what I'm trying to say, is you could learn algebra while you are taking a fun math class. Jump into a calc course and whatever you don't understand then back track to you get those basics, and then move onto the intersting material. Oh... google "MIT open course ware"... here I found the links for you:



http://ocw.mit.edu/OcwWeb/Physics/8-01Physics-IFall1999/CourseHome/index.htm

http://ocw.mit.edu/OcwWeb/Physics/8-02Electricity-and-MagnetismSpring2002/CourseHome/index.htm


Watch those videos... you will have a REALLY hard time finding a better physics lecturer then him. He's amazing.