Bemused at difficulty of Maths of 1st Year Physics.

This ended up a bit long and possibly a bit cynical but hopefully a little helpful.

First of all physics is a completely general subject in what is studied whereas engineering mainly confines its attention to man-made structures. An engineer will learn some specific maths skills during a degree , most of which they will never use again because engineering calculations are carried out for the most part using numerical models that have had an enormous and expensive effort made to test them in real-world situations. The expense is why we still use code developed for the Apollo program. Arguably the maths done by engineers is what is needed by engineering departments to set exams. It’s very nice to know that a V twelve engine has perfect balance up to a certain term in a Taylor series but is there a huge value in having to be examined on that vibration problem when training an engineer ? I don’t know but feel that other things could be done that have more value.

A physicist on the other hand is far more likely to have to generate their own mathematical models and come up with some kind of solution to them in particular circumstances although they will probably end up using computers as well. A physicist is far more likely to come across a problem that nobody else has looked at before.

Most of the physics type maths that you come across will be calculus based and a specific example is the second order differential equation. The most general of which is (in one notation and dimension)
f(x)y’’ + g(x)y’+h(x)y +p(x) + q=0. This is the most general form I can think of, of the Sturm–Liouville problem.

This is explored in various specific forms in maths courses as there is no universal analytical solution to the most general form. The vast bulk of equations you could write have no known analytical solution. Also you need to progress to 3D versions. In the end you will probably have to regurgitate solution techniques for your departments favourites such as Bessel’s equation each of which will be a simplified version of the Sturm–Liouville problem that some clever person such as D’Alembert, Euler, Bessel, Maxwell, Kelvin etc gave a name to because it could be applied to some specific, usually time dependent, situation. Things such as a vibrating membrane, a cooling body with a simple shape, a buckling cylinder, a travelling wave or whatever highly simplified situation they had dreamt up that fitted the solution method that had stumbled upon.

The new undergraduate will have to tackle functional notation and 3D vectors (both of which have recently been downgraded in A level maths). This maybe in an unfamiliar form presented by someone from the maths department who is resentful at being given a service course in the physics department to deal with. The 3D versions of the equations lead to the partial differential equations and the intimidating div, grad, curl and ultimately to Tensors and integral forms of the equations. BTW the div grad curl versions of Maxwell’s equations are considerably easier to deal with than Maxwell’s original forms.

Arguably, a great deal of this analysis could be dropped in favour of just using Mathematica etc. I don’t know whether that is a good idea because surely somebody should be working on fundamental blue sky maths. For example the “causal sets” Faye Dowker et al are working with (I don’t understand those BTW). Also, a physicist shouldn’t be just trusting what comes out of computers too much. Engineers can because they are applying computer solutions to the problems they were designed to solve.

You could do worse than taking a look at Div grad and curl and all that” and the Feynman lectures in physics. This will give you an overview of what the hell is going on in all those lectures.

As a finishing note, despite what Dr Sheldon Cooper might say, engineers are really useful and putting inquisitive physicists in charge of engineering situations is what leads to thing like Chernobyl.

Okay, so I have completed my first year in a undergrad Physics Degree at a Russel group Uni in the UK.
My results will be out in a couple days, but with all fingers crossed, after a hell of a lot of hard work I think I'll be on a first (I get £1000 for every year I get a 1st through a scholarship, hence why I care).
I'm slightly bemused and shocked by the difficulty of the maths that has been in the first year degree. I'm talking specifically about Vector/matrice calculus. By that I mean laplacian operators and flux operators in spherical and cylindrical and polar coordinates. Now there was a lot of tough maths but this stood out to me.
Asking around to engineers I know in 2nd year in different unis and the same, and reaching out to a maths/computer science guy finishing 2nd year, it appears none of them have done vector calculus to this level.
I'm slightly confused as to why the Maths in a Physics 1st year has transcended that of engineering 2nd year and a literal maths student 2nd year.
Perhaps I've just been unlucky and my sample size is small, if anyone can shed any light on this id really appreciate it.
Many thanks.