Check out this short tutorial on mechanics with calculus that I've written. It would make a good first introduction to physics[1]. There is another one for linear algebra[2].<p>In general I think programmers shouldn't fear the math and physics: yes it's hard to understand at first, but you can pick up things pretty fast. The ratio of "knowledge buzz" to effort is very good when you're learning physics. A symbolic computer algebra system like SymPy can be very helpful for "playing" with math expressions—here is a third tutorial on that[3].<p>_____<p>[1] <a href="https://minireference.com/static/tutorials/mech_in_7_pages.pdf" rel="nofollow">https://minireference.com/static/tutorials/mech_in_7_pages.p...</a><p>[2] <a href="https://minireference.com/static/tutorials/linear_algebra_in_4_pages.pdf" rel="nofollow">https://minireference.com/static/tutorials/linear_algebra_in...</a><p>[3] <a href="https://minireference.com/static/tutorials/sympy_tutorial.pdf" rel="nofollow">https://minireference.com/static/tutorials/sympy_tutorial.pd...</a>
I did math in the wrong order. I first was "exposed to it" as an engineer undergrad - didn't care at all (just wanted to code) and barely passed eng. math.<p>After working for two years, went back and did undergrad physics courses (including math for phys) and started to "get it".<p>Then went on to a PhD in general relativity - which is a LOT of math. I went back to software and most of it leaked away.<p>I am now trying to get it back (20 years later) - as I resurrect a Maple package for GR, grtensor, which I co-wrote as part of my PhD. Curved space is a beautiful thing.<p>Penrose's recent book reminded me of his wonderful "The Road to Reality: - which is a great tour of math leading into physics.
In my experience, it's a moving goal post. Physicists usually start off with the goal of explaining certain natural phenomena. Over the decades the specific parts of math this entails has evolved rapidly, as new phenomena become novel.<p>One might naively think that only "some" parts of math are useful for physics, but over the last 2-3 decades, many connections between physics and math have been discovered, that has brought the communities closer together, reversing the trend in the middle of the 20th century. For a very interesting take on the interplay between physics and math, by one of the masters of math/physics, see [1].<p>[1]: <a href="http://pauli.uni-muenster.de/~munsteg/arnold.html" rel="nofollow">http://pauli.uni-muenster.de/~munsteg/arnold.html</a><p>To quote an excerpt from Arnold's talk: " Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap. "
I studied physics in undergrad, and took a few grad courses. I also minored in math. The reason I dropped the math major to a minor was because of the amount of time mathematicians spent proving obvious things. Physics uses math, the way that a car mechanic uses a wrench without worrying about its atomic structure.<p>At the beginning of one particular interesting class on quantum mechanics, the professor wrote down a few shortcuts on the board: how a sum of a certain series collapses, which parameters can be ignored at low or high velocity, etc. It was glorious.<p>Also my big takeaway from this was that physicists really don't like non-linearity. If something cannot be described as linear, it will be described as a harmonic oscillator 99% of the time. The exception to this is statistical and simulation physics where you can do whatever you want and just look for emergent behavior.
The bullet points this article (and the one by Chad Orzel it links to) mentions are the common most used things day-to-day by basically anybody doing physics. However, I think stopping at that level of mathematical background would make reading a lot of the existing literature hard. Woit mentions complex analysis, but at least that usually comes up in mathematical methods classes, at least at the level needed to understand the arguments where it is used. Some math that I often found myself wanting more of includes differential geometry (I find physics introductions to tensor manipulation to be very heavy on mechanics and terrible on intuition for what you're actually doing) and functional analysis.<p>Both articles leave out numerical analysis or scientific computing, which I think is a huge gap. I certainly felt like my education left the impression that these things were way more straightforward than they actually are.
Those bullet points are the absolute baseline necessities that you need to have a shot at ever getting anywhere in physics.<p>On top of those, you will have to study a lot of other subject specific stuff or you will struggle. Physics education is way too light on math.
I did a double major in math and physics, many years ago. At the time, I didn't know if I was more interested in one than the other. I loved proofs. But I also loved the lab. So I just took all of the courses that were offered in both departments.<p>A few other people have mentioned that computation is missing from the list, and I agree. I was lucky that while studying my school subjects, I was also actively pursuing electronics and programming as hobbies, which drastically influenced my graduate studies in physics, and my subsequent career. But I didn't get that from my coursework.<p>I think that computation should be incorporated wherever possible in all of the coursework, all the way back to kindergarten. My rationale is simply that it expands the range of ideas that can be explored, and it's fun.
I was two years studying EE before I quit and transfer to CS program. I do bot regret it at all, but man am i grateful for hard math course and electromagnetics on EE. It was pretty painful, in terms of working hours and doing homeworks, but since then my brain just worked in terms of 2D/3D vision of geometry and space coordination in my head. Differential calculus, linear algebra, everything was piece of cake when I got to CS.
You need the math required to understand the language in which the current best physical theories are written. It's unknown as yet which mathematical objects will be required for their successors.
Feynman gave a whole lecture on this subject - the second or third lecture of the Messenger Lectures (<i>much</i> better timepass than night time TV show, second only to the Wizards Lectures and the five three seasons of The X-Files).<p>It seems that this is an ideal case for the Less Is More principle.