4 ms·

Under certain technical conditions, differentiation under the integral sign also works for Riemann integration (see Marsden & Hoffman). There's no need to devel

by r-zip 6y ago

Under certain technical conditions, differentiation under the integral sign also works for Riemann integration (see Marsden & Hoffman). There's no need to develop Lebesgue theory to demonstrate this technique in a calculus course, but uniform convergence must be understood.

judofyr 6y ago
Alternatively you can study physics and then you don't need to worry about these tiny details. I took a course on "mathematical methods in physics" which covered some complex analysis, and my math friends where shocked at how non-rigorous we were going through the theorems. Luckily for physicists these techniques tends to be valid because functions from the real world are well-behaved. For me personally it was so fun with a course where we did advanced mathematics for "practical" problems.
- sdenton4 6y ago
  It's all fun and games until you're trying to calculate trajectories over a Cantor set.
  - qubex 6y ago
    Or integrate over an interval of surreals.
- BeetleB 6y ago
  As someone who loved both math and physics, this was why I always found math a bit easier. Everything rests on a solid foundation and you can justify each step. When I got into higher physics, it was so riddled with intuitive arguments as opposed to rigor that I didn't fare so well. I'm sure one can find mathematical justifications for their methods, but it's not part of the curriculum, and almost none of the professors (in a top 10 physics school) knew them either.
  - TheOtherHobbes 6y ago
    Apart from the occasional Einstein and Newton every couple of centuries, physics seems to advance by throwing a semi-random selection of PhD dissertations at the real world and seeing if any of them happen to match experiment.
    - BeetleB 6y ago
      Case in point: Newton's work was not that rigorous. It was not till the 1800's that calculus was put on a firm foundation. Of course, things were all different back then.

by r-zip 6y ago

judofyr 6y ago

Alternatively you can study physics and then you don't need to worry about these tiny details. I took a course on "mathematical methods in physics" which covered some complex analysis, and my math friends where shocked at how non-rigorous we were going through the theorems. Luckily for physicists these techniques tends to be valid because functions from the real world are well-behaved. For me personally it was so fun with a course where we did advanced mathematics for "practical" problems.

sdenton4 6y ago
It's all fun and games until you're trying to calculate trajectories over a Cantor set.
- qubex 6y ago
  Or integrate over an interval of surreals.
BeetleB 6y ago
As someone who loved both math and physics, this was why I always found math a bit easier. Everything rests on a solid foundation and you can justify each step. When I got into higher physics, it was so riddled with intuitive arguments as opposed to rigor that I didn't fare so well. I'm sure one can find mathematical justifications for their methods, but it's not part of the curriculum, and almost none of the professors (in a top 10 physics school) knew them either.
- TheOtherHobbes 6y ago
  Apart from the occasional Einstein and Newton every couple of centuries, physics seems to advance by throwing a semi-random selection of PhD dissertations at the real world and seeing if any of them happen to match experiment.
  - BeetleB 6y ago
    Case in point: Newton's work was not that rigorous. It was not till the 1800's that calculus was put on a firm foundation. Of course, things were all different back then.