Useful maths?

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Frh
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Useful maths?

Postby Frh » Mon Jun 11, 2007 12:33 am UTC

Aloha, I have questions for you good people!

What areas of maths would you suggest someone in his 2nd year of a physics degree, with a lot of time on his hands, studies?

Also, if I were to attempt learning a programming language, what would be most useful to a physicist? And more importantly, most free!

Frh

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Postby Gordon » Mon Jun 11, 2007 12:42 am UTC

Ain't nothing free in life!

Hm if you're in physics you could look into studying the maths behind fractals. They're pretty cool. My problem was I always like doing the cool things though.

Since all you goobers seem to like wikipedia so much:
http://en.wikipedia.org/wiki/Fractals
Though there isn't any math on that page from what I glanced at. If you're in school you can take a look through the course book and see if any courses cover fractals... probably either a theoretical physics course or an algebra/geometry course.
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Postby SpitValve » Mon Jun 11, 2007 1:35 am UTC

Do lots of calculus and differential equations: they're the basis for most of physics.

Also do linear algebra if you want to do quantum mechanics.

If you want to ever understand general relativity, you need a lot of mathematics. A good handle on set theory and calculus should get you off to an ok start when you do the differential geometry behind general relativity.


For programming, I'd recommend Fortran and C. You can get free compilers for both: g77 or g90 or g95 for Fortran depending on what version you want, and gcc for C. C is a more versatile language, but Fortran - specifically, the 1977 version of Fortran - is pretty much the standard for science programming.

It might also be worth playing with C++ or Java just to get your head around objects as well.

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Postby ArmonSore » Mon Jun 11, 2007 1:48 am UTC

Imaginary numbers are also very useful to a Physicist. An understanding of Fourier Series wouldn't hurt either(in fact, any class on numerical analysis (a.k.a. making useful approximations) is useful).

Well, those are some of the classes I'm taking/working on pre-requisites to take.
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Postby SpitValve » Mon Jun 11, 2007 3:18 am UTC

Yes! Fourier transforms are bloody everywhere, you need those.

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Postby adlaiff6 » Mon Jun 11, 2007 4:16 am UTC

You could learn C or Fortran, but Mathematica would be pretty good as well or instead.
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Postby SpitValve » Mon Jun 11, 2007 4:19 am UTC

adlaiff6 wrote:You could learn C or Fortran, but Mathematica would be pretty good as well or instead.


People use matlab etc too, but for serious numerical work you need the speed of Fortran, I reckon.

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Postby bippy » Mon Jun 11, 2007 4:58 am UTC

A second year physics student should get a copy of Mathematical Methods in the Physical Sciences by Boas and start a'workin.

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Postby skeptical scientist » Mon Jun 11, 2007 5:29 am UTC

You generally want to take as much calculus/multivariable calculus/differential equations/analysis/geometric analysis as you can. Linear algebra is important insomuch as you need it to do those; more abstract algebra is probably less important than the analysis side of things, but could be useful for some things in quantum mechanics and perhaps string theory - I don't really know how the stranger varieties of algebra get used, but I've heard that they do.

Here is an attempt at a rough breakdown of what's used where, and why you need it, in the format (Math topic: where it's used in physics). This isn't so much intended to tell you what you should be learning now, but rather what math is out there, and how it is used in physics. Right now, I would learn linear algebra and multivariable calculus if you haven't already, and otherwise real analysis.

Basic stuff, important to do anything later:
Calculus: Everywhere
Linear Algebra: necessary to understand anything involving more than one variable
Multivariable calculus: everywhere
Real analysis: necessary to understand any of the more advanced analysis, and why stuff in calculus is true.

If you just intend to get an undergraduate degree, and then get a job, this much math is probably fine, although you may want to take a differential equations class and/or a complex analysis class. If you intend on performing postgraduate study and research, you'll need quite a bit more.

More advanced stuff (you may need some or all of these depending on what you want to do with your physics degree):
Complex analysis: anything having to do with waves and oscillations (in other words, all of quantum mechanics, and quite a lot of classical mechanics besides)
Differential equations: everywhere
Topology: Important for string theory, sometimes used elsewhere
Calculus on manifolds (you may not get to this until graduate school, if you end up going that route): general relativity, string theory
Riemannian geometry (you almost certainly won't get to this until graduate school, if you end up going that route): general relativity
Functional analysis (you almost certainly won't get to this until graduate school, if you end up going that route):
-Fourier transforms: comes up everywhere when performing/designing/interpreting data from experiments, also important for quantum mechanics
-Distributions: quantum mechanics - you need to deal with these any time you have a probability distribution which isn't smooth; for example, when you know the starting location of a particle, and want to use QM to see how it propagates.

Several of the more advanced topics are taught in the physics classes where they are used, alongside the physics. However, if you intend on going to graduate school, I highly recommend taking a math class where you see it as well, because a. you can devote a lot more time to it when you're not trying to learn physics in the same class at the same time, and b. in my experience, the math classes are a lot better at explaining what is actually going on, while the physics classes tend to be more focused on how to perform calculations, without actually explaining anything. I remember being told in quantum mechanics to just pretend distributions were functions and I just wanted to scream "what the fuck, this makes no sense!" because you're performing operations which only make sense when applied to continuous functions, but the objects you are working with are neither continuous nor functions.

In general, how far you go in physics is heavily dependent on the amount of math you know, so depending on your post-undergraduate plans, the more you can learn the better.
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Postby SpitValve » Mon Jun 11, 2007 6:20 am UTC

ooh and Group Theory for quantum, but that sorta comes under linear algebra I guess?

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Postby xooll » Mon Jun 11, 2007 6:26 am UTC

bippy wrote:A second year physics student should get a copy of Mathematical Methods in the Physical Sciences by Boas and start a'workin.

Yup.
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Postby Zohar » Mon Jun 11, 2007 7:26 am UTC

I study both physics and math in the university. Calculus is important, certainly. And it would be nice to know when you can really change a sum into an integral or change the order of sums etc. But that usually helps in actually solving problems.

Without a doubt, the most helpful and important subject that I learned is linear algebra. Also, Hilbert spaces. Both are incredibly important, not only to quantum mechanics but also to a lot of earlier stuff (waves, mechanics and so on).

That physics is taught (at least in our university) without proper knowledge of linear algebra is a crime against science. Well, maybe not a crime but "very bad".
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Postby djn » Mon Jun 11, 2007 9:32 am UTC

SpitValve wrote:People use matlab etc too, but for serious numerical work you need the speed of Fortran, I reckon.


Incidentally, I've just spent half a year studying, among other things, how to integrate fortran and python. Basically: It's easy.
Create fortran function in file, run f2py (fantastic tool) on it, and you get a python module with the functions in the fortran file. If you manage to arrange your data so it goes into and out of fortran without any copying being done, it's almost as fast as using straight fortran, but you get the niceties of python around it.

Oh, and I suggest python. The numPy (or Numarray, or Numeric) module makes doing things with large arrays fast(er), the language itself is neat, and you can farm out to fortran (or C, which is a touch more complicated) if you need more speed.

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Postby evilbeanfiend » Mon Jun 11, 2007 10:29 am UTC

you also need a small amount of stats for thermodynamics.
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Postby rhino » Mon Jun 11, 2007 12:31 pm UTC

I agree 90% with skeptical's list of topics, except that I think differential equations are *really* important and you should definitely learn about them at least a little bit. Just about every dynamical system is governed by a differential equation. They are advanced though in that you can't do them without solid grounding in calculus.

I also think analysis isn't particularly important for physicists- proving how all the calculus tools work isn't necessary or even particularly useful for being able to apply them. I'd recommend it to anyone interested in maths for its own sake, but since you seem to be concerned about applications I wouldn't prioritise it.

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Postby skeptical scientist » Mon Jun 11, 2007 3:38 pm UTC

rhino wrote:I agree 90% with skeptical's list of topics, except that I think differential equations are *really* important and you should definitely learn about them at least a little bit. Just about every dynamical system is governed by a differential equation. They are advanced though in that you can't do them without solid grounding in calculus.

I also think analysis isn't particularly important for physicists- proving how all the calculus tools work isn't necessary or even particularly useful for being able to apply them. I'd recommend it to anyone interested in maths for its own sake, but since you seem to be concerned about applications I wouldn't prioritise it.

I was assuming that a bit of differential equations would be taught in undergraduate calculus and/or physics, so you could at least get the idea without taking a class on differential equations specifically. In my experience, differential equations classes aren't that interesting as they mostly just go over a laundry list of different techniques for solving different types of differential equations. On the other hand, they can be very useful.
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Postby QuantumTroll » Tue Jun 12, 2007 8:20 pm UTC

I'm a comp sci major who went into physics. I just have this to say: For the love of God learn to program!!!

FORTRAN is great, and easier than C, although learning C will allow you to learn any language in a heartbeat. It's the hardest, IMO, if only because of stupid memory management issues.

Python has a great science library. Check it out if you're interested in just tooling around and being productive...

Math-wise, you can't go wrong if you polish up your differential equations. An impoverished math background is my Achilles Heel in physics classes.

Good luck!

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Postby Vaniver » Tue Jun 12, 2007 8:31 pm UTC

I use Fortran exclusively for my physics work (as in, what I get paid for). It's surprisingly easy to pick up and use (although, to be fair, I also have a programming background).
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Postby MathBOB » Tue Jun 12, 2007 9:31 pm UTC

What shouldn't you study? I'm a math major who wants to go to grad school for mathematical physics (therefore I am a total nerd), and I haven't really found much math that hasn't worked it's way into physics.

Rigorous analysis courses get worked into classical mechanics, and real analysis (measure theory) as well as non-standard analysis has found a home in some areas of quantum mechanics.
Linear algebra, besides being a useful tool, is basically the entire basis of quantum mechanics. Besides that, it provides a jump-start into...regular algebra? Group, ring, and field theory are central to modern particle physics and all of the quirky supertheories that are trying to establish themselves.
Topology and differential geometry are key to electromagnetism, relativity, and the aforementioned supertheories. Differential geometry has also taken a strange role in classical mechanics and theoretical physics, as determinations of Lagrangians and the least action principle can lead to interesting geometric definitions.

And of course, a physics major who doesn't take Fourier and complex analysis should be shot on the spot. If you're not really all that interested in math, you should still take:
Multivariable analysis
Linear algebra
Fourier analysis
Complex analysis
Differential equations [Edit]
At least. Or rather, at least learn all those things.
Last edited by MathBOB on Wed Jun 13, 2007 12:27 am UTC, edited 1 time in total.

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Postby QuantumTroll » Tue Jun 12, 2007 10:53 pm UTC

While there's an amazing amount of math that is used in physics, the maths that help a student in their 2nd year is more limited. Calculus and related mathematics are used extensively. Group theory... not so much. That's more for Quantum Field Theory and suchlike, as I understand it.

Knowing your way around algebra/trig is, of course, a must, but I don't think the sort of person who asks about what math to study in their free time has any trouble with that stuff...

On a more academic note, maybe it'd be worthwhile to practice mental math. That helps improve physical intuition and test-taking.

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Postby Woxor » Tue Jun 12, 2007 10:59 pm UTC

In engineering, we mostly used algebra (like, basic algebra, not abstract algebra), but it could still be really hard. You've got to be able to manipulate data when you've got ten (usually non-linear) equations with ten unknowns, except that maybe some of the "equations" are approximations that need to be augmented by other data, and many times you'll have a table or graph instead of an equation, etc. I don't know how much of that would apply to abstract physics, though.

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Postby MathBOB » Wed Jun 13, 2007 12:27 am UTC

I maintain that the requisites for a physics major would be:
Multivariable calculus (i.e. Stokes' theorems)
Linear Algebra
Fourier analysis
Differential equations
And of course, anything that comes before then. You need multivar to do classical mech and quantum mech, linear algebra and Fourier analysis to do quantum mech, and differential equations...for everything. Sorry, I messed up and said "differential geometry" the first time.

You should keep in mind that I'm more geared towards mathematical physics (or, if you prefer, "mathematically rigorous physics") at a school where the physics department is geared toward that. They expect you to have taken at least this much before finishing your second year of physics, and these plus at least 1 higher-level math class is required for gradumation. I'm just a *tad* biased towards math.

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Postby adlaiff6 » Wed Jun 13, 2007 8:07 am UTC

SpitValve wrote:
adlaiff6 wrote:You could learn C or Fortran, but Mathematica would be pretty good as well or instead.


People use matlab etc too, but for serious numerical work you need the speed of Fortran, I reckon.


Right. Mathematica, Maple, MATLAB, and whatever else there is are pretty much equivalent. I said Mathematica because I happen to know it best.

C will be good to get you in to programming (and is nice if you just have something quick you want to bang out---which is more common in the sciences than actually writing an application for other people to use). Fortran is much faster, and if you're doing real big maths, you'll want to bring out the real big guns.
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Postby Yakk » Wed Jun 13, 2007 5:29 pm UTC

adlaiff6 wrote:
SpitValve wrote:
adlaiff6 wrote:You could learn C or Fortran, but Mathematica would be pretty good as well or instead.


People use matlab etc too, but for serious numerical work you need the speed of Fortran, I reckon.


Right. Mathematica, Maple, MATLAB, and whatever else there is are pretty much equivalent. I said Mathematica because I happen to know it best.

C will be good to get you in to programming (and is nice if you just have something quick you want to bang out---which is more common in the sciences than actually writing an application for other people to use). Fortran is much faster, and if you're doing real big maths, you'll want to bring out the real big guns.


*blink* -- I always figured MATLAB wasn't nearly the same beast as Maple.

MATLAB is scientific computation, while Maple is symbolic mathematics...

Have they intersected since?

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Postby dumbclown » Wed Jun 13, 2007 7:52 pm UTC

MATLAB and Maple are still different.

You could take a course in compiler design. It will teach you a few amount about programming techniques.

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Postby Yakk » Fri Jun 15, 2007 8:20 pm UTC

I recently discovered a set of open-source math tools that attempt to replace matlab, maple and that statistics package of evil (tm).

Can you guess which of the three I liked least?

There are some neat ASL*s in C++ that do interval-based mathematics and the like.

Python is neat to use, because they don't provide you with a bazooka aimed at your feet, armed & loaded, then place your hand on the trigger, unlike C++. Of course, if you never do bazooka programming, you never do learn how to use a bazooka. :)

Fractals look pretty, but from what I've heard most of the math is pretty pedestrian up to the point of "we have no clue" hits. You could learn about non-integer dimensions, which is neat & slightly mind-blowing.

* application specific languages. You write in C++ a set of objects that create a non-C++-esque syntax in C++. An example of this is boost::lambda.

Calculus on manifolds (you may not get to this until graduate school, if you end up going that route): general relativity, string theory
Riemannian geometry (you almost certainly won't get to this until graduate school, if you end up going that route): general relativity
Functional analysis (you almost certainly won't get to this until graduate school, if you end up going that route):


And that is why I liked my undergraduate program. We made the graduate students cry. :)

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Postby MathBOB » Fri Jun 22, 2007 3:39 pm UTC

Yakk wrote:
Calculus on manifolds (you may not get to this until graduate school, if you end up going that route): general relativity, string theory
Riemannian geometry (you almost certainly won't get to this until graduate school, if you end up going that route): general relativity
Functional analysis (you almost certainly won't get to this until graduate school, if you end up going that route):


And that is why I liked my undergraduate program. We made the graduate students cry. :)


(Honestly, I'm confused as to what you mean by this...I don't understand why the grad students would cry. Sorry!)

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Postby Yakk » Fri Jun 22, 2007 6:57 pm UTC

Oh -- we had undergrad courses (mostly 4th) that taught grad-level material. So grad students would show up and be taking courses with undergrads, and often have less preparation for the material.

It was amusing.

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Postby MathBOB » Fri Jun 22, 2007 11:14 pm UTC

Ah, I can see that. At my school, we have a strange system (that I believe clearly shows the superiority of mathematics): undergrad math courses count towards the physics degree, but only graduate physics courses can count towards the math degree. In the math courses, we have the same situation you do. Our "introduction to" grad math courses are anything but an introduction, so grad students end up taking the undergrad classes.

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Postby skeptical scientist » Sat Jun 23, 2007 12:48 am UTC

Where did you go Yakk? Sounds kind of like my university, although only very few of the grad students end up taking undergrad courses. But the undergrads here are top notch, and the ones who are taking the grad classes do very well; often better than many of the graduate students.
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Postby Frh » Tue Jul 03, 2007 7:42 pm UTC

Thanks for the tips guys! Contacted one of my lecturers in addition to asking here, got basically the same advice ;). I purchased Mathematical Methods in the Physical Sciences, and have been working through that fairly steadily.

I now have a new question, regarding programming! Where do you find problems to solve ? I tried to read various manuals to get the syntax down (Python by the way), but without actually having some problem to solve, I don't know what to do. Thus far I have just made a program to solve sequences and series for me... it was not very taxing! I tried to make some stuff up my self, but as I don't know the language I don't know what it can do, bit of a catch 22!

Thanks again,

Frh

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Postby aoanla » Tue Jul 03, 2007 8:40 pm UTC

Frh wrote:
I now have a new question, regarding programming! Where do you find problems to solve ? I tried to read various manuals to get the syntax down (Python by the way), but without actually having some problem to solve, I don't know what to do. Thus far I have just made a program to solve sequences and series for me... it was not very taxing! I tried to make some stuff up my self, but as I don't know the language I don't know what it can do, bit of a catch 22!

Thanks again,

Frh


By definition, any (Turing-complete) programming language can solve any computable problem. So, really, pick any soluble problem you like.

I suggest writing a Newton-Raphson iterative root finder, because you can trivially modify it to draw cool fractals as well.
(Since you're using Python, I assume you have gotten hold of NumPy and SciPy? You might also like to pick up matplotlib for the graphing tools.)

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Postby ks_physicist » Wed Jul 04, 2007 11:49 am UTC

SpitValve wrote:
adlaiff6 wrote:You could learn C or Fortran, but Mathematica would be pretty good as well or instead.


People use matlab etc too, but for serious numerical work you need the speed of Fortran, I reckon.


A fellow grad student and I both used Matlab to do some simple electromagnetic calculations and modeling (independent from each other, he did magnetic field modeling and I did electrostatics).

At the time I started, I discovered that a million point matrix was a little slow. With modern computers, I can do many millions of points in a reasonable time (~ 1 coffee), and considerably more points in not too bad of a time period (1 day to 1 week).

We decided that if we want to improve the performance of our programs, we would ideally either convert entirely to Fortran, or write modules in Fortran that could be called by Matlab. We went so far as to think of using parallel computing clusters, but without funding we shelved that and just finished our Matlab simulations.


As previously mentioned, fractal mathematics is fascinating, and if you're looking for mental stimulation it's a good way to go. If you're looking for more directly beneficial mathematics, I can almost always recommend studying more calculus, diff EQ, and linear algebra.

Series analysis methods, especially Binomial/Taylor/MacLaurin and Fourier, are frequently used. Good to study up on those if you haven't already.

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Postby ks_physicist » Wed Jul 04, 2007 12:01 pm UTC

bippy wrote:A second year physics student should get a copy of Mathematical Methods in the Physical Sciences by Boas and start a'workin.


+ a lot.

One of my favorite mathematical texts, it was used in my theoretical physics course. I will never part with it.

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Postby ks_physicist » Wed Jul 04, 2007 12:06 pm UTC

Yakk wrote:*blink* -- I always figured MATLAB wasn't nearly the same beast as Maple.

MATLAB is scientific computation, while Maple is symbolic mathematics...

Have they intersected since?


Matlab has included the Maple kernel in its "Symbolic Math Toolbox" (optional) for at least a decade.

But yes, the focus of each remains distinct.

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Postby Yakk » Wed Jul 04, 2007 2:18 pm UTC

skeptical scientist wrote:Where did you go Yakk? Sounds kind of like my university, although only very few of the grad students end up taking undergrad courses. But the undergrads here are top notch, and the ones who are taking the grad classes do very well; often better than many of the graduate students.


U(W). Brackets signify.

To be fair, it is a mainly undergraduate school. Analysis grad students mostly end up taking the entire 4th year set of analysis undergrad courses to bring them up to speed, as a concrete example.

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Postby MathBOB » Thu Jul 05, 2007 5:38 pm UTC

Again, sounds like math at my school. I swear it's like they detest the grad students. "Introduction to" graduate courses are anything but, so you get a bunch of 1st-years taking the undergrad analysis and algebra courses, since they refuse to offer them at the grad level. There are almost no grad math courses at all (at least, in pure math).


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