As (failed) physics major, it sounds so strange to hear someone claim to be good at math but not physics. To me physics is 98% math. It's a few empirical facts from experiments and then bucketloads of math.
For example, picking a nice reference frame in simple mechanics problems is something that a physical intuition is good for. Same with spotting symmetries in an EM problem. Also, in physics you need to have a good grasp of what to ignore because they have only a small effect on the solution or because it operates on a different scale (fringing effects, transient solutions in ODEs), which often relies on a very hand-wavey type of reasoning.
Essentially, physical intuition often does not map to mathematical intuition
I have not taken enough math to actually speak for them; this is mostly gleaned from talking with math major friends and my own speculations.
> Also, in physics you need to have a good grasp of what to ignore because they have only a small effect on the solution or because it operates on a different scale (fringing effects, transient solutions in ODEs), which often relies on a very hand-wavey type of reasoning.
Physics is often taught in an exceedingly vague and hand-wavey way. Physics students have to learn to ignore that nagging feeling that something is not quite right. This is impossible for a mathematician. A mathematician wants to cleanly separate the math from the problem that is being solved using math. The problem specification consists of a list of assumptions, and the solution of the problem consists of 100% rock solid math. Physicists weave the two together, so that in the end it's often not clear what is actually being assumed. Furthermore, it's usually not explained based on which experiments those assumptions are justified. A counterexample is special relativity. There it's clearly assumed that the speed of light is constant, and the experiments on which that assumption is based are explained, and from there it's mostly logical deduction. In other topics that is sadly not the case. I would love a physics education where you start with the experiments and work from there, instead of saying "Bam! Here are Maxwell's differential equations. Now deduce things from that based on hand-wavey arguments". I don't mean having the students perform the experiments themselves, just describe what somebody else did and what the results were, and why that led people to believe that the laws of physics are as they are. To make time for that, we should remove the endless by hand solving of special cases of special cases. We live in the 21st century. Instead use numerical methods everywhere, which easily tackle the general case. Got n electrons with initial positions and initial velocities, and you want to see what happens? No problem.
/rant
Ironically though, I ended up drifting into the EECS department to do applied physics and I've found an environment much more similar to mathematics (and to the experiment-based approach you advocate) than to the physics department -- when you're trying to build systems rather than just solve problems, you can't just wave your hands. Instead, you have to pick apart your assumptions and figure out why you can ignore certain things and not others.
It's odd that you chose that example, because there is a rich history leading up to Maxwell doing his work, then Hertz verifying it, then Heaviside refactoring the notation.
Perhaps the deeper issue is that there are only so many lecture hours in a semester for adding these details, and you gotta start somewhere.
These things helped me get by.
Hah, I did the same :-) ... unfortunately I also forgot the conventional names, and just used 's' (for speed) instead of 'v', and 'd' (for distance) instead of 's'. My teacher did not object the derivation at all, but told me that the results are wrong, because of the letters used ...
Well, point taken - now, I try to pay extra attention and memorize idiosyncrasies like naming conventions ... it saves time when communicating with others.
You should have been exposed to physical chemistry or computational chemistry or maybe even quantum chemistry or analytical chemistry, if your are mathematically inclined.
...but unfortunately they are considered very advanced topics in most learning institutions, even if one could start with them from the very beginning, instead of "classical chemistry". And more unfortunately, after the tedium of "classical chemistry", what you are presented with next are very boring aspects of "organic chemistry" or "biochemistry".
The chemistry department in the university I studied at required 4 courses of physical chemistry. The introductory course was bad enough - the course had mandatory practice sessions, where assistants were at hand to aid with the supposedly trickier bits. Each session was 1h45m straight.
5 weeks in, there was a supposedly simple exercise. When nobody at the class got even past the initial hurdles in the first 15 minutes, the assistant decided to show how it's done. He failed to finish the calculations in the remaining 90 minutes ... and he knew how the steps went.
Eventually I changed my major from chemistry to CS.
EDIT: btw, the assistant in question was a post-grad so lack of domain knowledge was not the reason.
In my view, the starting point for physics is a deep curiosity about how things work. But I'm not sure that aspect of it is ever taught. Rather, it's assumed that good physics students arrive at college, having developed that instinct on their own as kids -- taking things apart, breaking things, asking questions, maybe having curious parents.
Instead, the emphasis in teaching physics is almost purely on the math. Certainly, what defines physics as a unique discipline is the interest in studying problems that lend themselves to mathematical analysis. Solving the textbook problems involves identifying the equations corresponding to the wording of the problem, then solving the equations. That's a skill, but it's not really physics.
I got through my physics courses on the strength of my math skills, but was extremely fortunate to have picked up the empirical half of physics on my own, through my hobbies, and from the curiosity about nature that my parents encouraged. But if someone lacks that background, I could see them being good at math, and maybe getting a good way through school physics, but never really getting physics as an end unto itself.
it wasn't enough to tell her "well it still gives us the valid results" - she had to be able to interpret every portion of an expression and then make intuitive sense of it.
at first i was annoyed with this habit of hers, but eventually i started doing it to. for a lot of really math-minded people, hand waviness is anathema, and physics was full of "2 + 2 = 5 for sufficiently large values of 2, so we'll just assume that to make things easier"
I've tried to teach friends with that hangup and found it difficult. Those people weren't good at math, though. I feel like with mathematicians, you should just be able to explain with limits.
Assuming the limit exists without actually proving the sequence converges is going to produce total junk. Jaynes spent a good portion of Probability Theory illustrating this.
Each of those is just a definition and a handful (literally like 5 or 6) of axioms. If I memorized the axioms and a few key theorems I could usually derive everything else in the homework and on the exams. Most mathematical objects have very similar structure and the rest is just maps (morphisms) between them.
> Oh and those greek letters.
That's actually a very valid point. It took me a few years after graduating to realize how much myself (and I suspect many other students) are hampered by not knowing the notation well enough. Most people just assume you (and they) know what it means and never think about the actual definition and ambiguities. Even not being able to pronounce Greek letters definitely makes you less able (or at least less confident) to reason with them.
> Calc 2 techniques of integration and then DiffEq seemed like a lot of special cases that had unique solutions we had to remember.
That's the applied math part of calculus. The pure side is just "here's a compact/open/whatever set, prove for any point in it this property holds." Then you find out no one cares about either and it's all just numerical algorithms.
Is there a book covering this topic that you can recommend?
I recall that a physics chapter would start by "we take the Maxwell equations as [complex formulae]..." and for the life of me I wasn't able to understand them, or see where the teacher took that from.
On the other hand, maths seemed more logical, and even now - almost 15 years later - I can correctly recall my undergraduate maths classes, because I have them so well ingrained in me, because I could understand how various ideas connected to each other.
some of this is maybe inherent to the aim of physics, some of it is just physics machismo culture, and could be better, imo.
(i was a physics and math major at oregon.)
I think the best advances in both math and physics are made with people with an excellent grasp of when to move forward with existing constructs, and when to start shaping new ones...
Math in physics (at least most of the physics I have done, which covers classical 19th century stuff mostly) is just a tool, if you can take a shortcut, take it! If you can approximate and cheat, do it! What matters in physics is the path from observation to modeling. Math is but a tool.
Now I agree that the math gets rather solid, but it's nothing compared to real math at an equivalent level.
I think my experience is particular because in France where I studied, you study both in parallel very intensively. So the math in physics always kind of seems trivial to you... But still my best physics teacher taught me that it was way more about the "feel and model" than the "exactly prove" that mathematics consists in.
Gosh I miss those days :)
There is a lot to math.