10 ms·

That is only if your allowing yourself to take the numerical methods as a black box. If you want to be sure that your method is okay, you prove error bounds as

by Cobord 8y ago

That is only if your allowing yourself to take the numerical methods as a black box. If you want to be sure that your method is okay, you prove error bounds as you take limits of 0 mesh size. Otherwise it is just building tables of black magic for what schema to use when.

jstewartmobile 8y ago
That's kind of where I was going with this. Instead of burning all of that time on symbolic differentiation, dig-down into numerical methods ASAP so students can get a feel for all of the related "gotchas"--of which there are many... edit: IMHO, many of those "gotchas" are much more interesting than the fundamentals of calculus.
- Cobord 8y ago
  Consider symplectic integrators. You would never come up with them or realize the problem of energy drift if you hadn't first paid attention to the fundamentals of the geometry and calculus underlying the problem. This is just my favorite example, but it illustrates how understanding the fundamentals also explains the gotchas. Just getting a feel for them through experience is again just black magic by building up a table of what to use when without the generalizing principle behind it.
  - jstewartmobile 8y ago
    Never said to do away with the fundamentals. Will say that most symbolic differentiation and integration (which is a big chunk of the coursework) is not fundamental as much as it is fruitless busywork. Even so, I spent a year of my time--and God knows how much of other people's money--grinding out the mathematical equivalent of crossword puzzles so I could get my job certificate--just like every other engineer. Use that same time to apply the fundamentals to numerical methods, and you get to go in far more interesting directions--like symplectic integrals, or chaos theory.
    - BeetleB 8y ago
      >Never said to do away with the fundamentals. Will say that most symbolic differentiation and integration (which is a big chunk of the coursework) is not fundamental as much as it is fruitless busywork. It's no more busywork than being able to multiply two single digit numbers in your head. Whether it's useful to your job really depends on the job. I had a job once in the engineering industry. When we were in meetings discussing projects, if you could not do those types of analyses (e.g. asymptotic behavior of certain Calc II type integrals) in your head, you would not know what's going on. Sure, everyone could explicitly show all the steps for your benefit, but you'd be slowing everyone down.

by Cobord 8y ago

jstewartmobile 8y ago

That's kind of where I was going with this. Instead of burning all of that time on symbolic differentiation, dig-down into numerical methods ASAP so students can get a feel for all of the related "gotchas"--of which there are many... edit: IMHO, many of those "gotchas" are much more interesting than the fundamentals of calculus.

Cobord 8y ago
Consider symplectic integrators. You would never come up with them or realize the problem of energy drift if you hadn't first paid attention to the fundamentals of the geometry and calculus underlying the problem. This is just my favorite example, but it illustrates how understanding the fundamentals also explains the gotchas. Just getting a feel for them through experience is again just black magic by building up a table of what to use when without the generalizing principle behind it.
- jstewartmobile 8y ago
  Never said to do away with the fundamentals. Will say that most symbolic differentiation and integration (which is a big chunk of the coursework) is not fundamental as much as it is fruitless busywork. Even so, I spent a year of my time--and God knows how much of other people's money--grinding out the mathematical equivalent of crossword puzzles so I could get my job certificate--just like every other engineer. Use that same time to apply the fundamentals to numerical methods, and you get to go in far more interesting directions--like symplectic integrals, or chaos theory.
  - BeetleB 8y ago
    >Never said to do away with the fundamentals. Will say that most symbolic differentiation and integration (which is a big chunk of the coursework) is not fundamental as much as it is fruitless busywork. It's no more busywork than being able to multiply two single digit numbers in your head. Whether it's useful to your job really depends on the job. I had a job once in the engineering industry. When we were in meetings discussing projects, if you could not do those types of analyses (e.g. asymptotic behavior of certain Calc II type integrals) in your head, you would not know what's going on. Sure, everyone could explicitly show all the steps for your benefit, but you'd be slowing everyone down.