> What is it like to understand advanced mathematics?
It's really nice! Get to understand
a lot of stuff.
Some of the advanced math is really
powerful for applications.
And, can look back at the math
saw in physics where wondered if the
physics profs really understood the
math and conclude, right, they
didn't or at least not very well.
E.g., recently I saw a physics lecture
where they explained more than once
that a quantum mechanics wave
function was differentiable
and "also continuous" as if
there was some question, doubt,
choice, or chance otherwise. Of COURSE
it is continuous! Every differentiable
function is continuous!
There's a lot more on why it's
nice to understand the advanced
math!
Under who's definition? The Heaviside step function is discontinuous, but it's derivative is usually considered to be the Kronecker delta function. Both of these are used extensively in physics and engineering.
No, it is just that physicists means something different than mathematicians when they use the word "function". Just assume that they always are distributions that are evaluated with dirac deltas and everything makes sense.
They most likely said "The wave function is differentiable and its derivative is continuous", which isn't obvious. If you think that it sounded strange you should have asked during the lecture and they would have explained. Now instead you just assumed that you knew better and made a fool of yourself...
Also this isn't advanced mathematics, this is something most who are interested learns in high-school and I can assure you that most physics professors are aware of this fact.
No, they said what I wrote they said. It was at YouTube, from MIT, from a course in quantum mechanics.
Uh, I do understand this stuff: E.g., given a positive integer n, the set of real numbers R, the usual topology for R^n, a set C a subset of R^n closed in that topology, there exists a function
f: R^n --> R
so that for x in R^n, f(x) is 0 for x in C, f(x) > 0 for x not in C, and f infinitely differentiable. E.g., for n = 1, the result applies for set C a Cantor set or a Cantor set of positive measure. For n > 2, the result applies for a sample path of Brownian motion or the Mandelbrot set.
When I was a grad student, I discovered and proved this result. Later I published the result in JOTA.
A bigger problem is the difference between Hermitian and self adjoint, a lot of physics professors don't understand that.
Or for example, this famous dialogue between Geoffrey Chew and Arthur Wightman from the 1960s bootstrap fad:
Wightman asked Chew: why assume from the start that the S-matrix was analytic? Why not try to derive it from simpler principles? Chew replied that "everything in physics is smooth". Wightman asked about smooth functions that aren't analytic. Chew thought a moment and replied that there weren't any.
By the way, the reason they said that was no doubt to motivate the students to choose correct boundary conditions for some Sturm-Liouville problem (as the previous poster sort of alluded to)
It's really nice! Get to understand a lot of stuff.
Some of the advanced math is really powerful for applications.
And, can look back at the math saw in physics where wondered if the physics profs really understood the math and conclude, right, they didn't or at least not very well.
E.g., recently I saw a physics lecture where they explained more than once that a quantum mechanics wave function was differentiable and "also continuous" as if there was some question, doubt, choice, or chance otherwise. Of COURSE it is continuous! Every differentiable function is continuous!
There's a lot more on why it's nice to understand the advanced math!