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Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

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Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#71

Post by ragulpr »

codewithcheese wrote: post #55
burnerOh2125 wrote: post #31

Interesting, I'm a phd-drop out in computational biology, working as a data science consultant: I use mathematics including multi-dimensional statistics, linear algebra, and calculus everyday. Being self-taught, I'm very self conscience about the math I don't know, but so far, not knowing differential equations doesn't seem to have hurt me. I actually just ran into a problem that uses Hamiltonian dynamics, so maybe I…

Hi I am interesting your advice on which math to learn if you have a spare moment to provide it. I too am starting to teach my self the requirements of data science and am also self conscious of the math I don't know. I am very excited about what is now possible with machine learning and deep learning as I believe it will become increasingly necessary for developers to stay relevant. Could you comment in more detail…

* Solid linear algebra

* Probability theory

* Statistics (bayesian if possible)

* Optimization (this is less important but extremely useful)

If you know the mechanics of multivariate calculus you'll be fine learning the above. The course that personally have had most payoff was functional analysis. Purely theoretical course that will give you no practical skills and at first glance seems unrelated to ML but it (subtly) gave me a much deeper understanding of what ML is all about.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#72

Post by wolfgke »

CalChris wrote: post #32
stephengillie wrote: post #2

Please teach calculus before teaching trigonometry. There's no prerequisite to learn trig first, and forcing people to learn trig-calc excites many mathophiles but is a major turn off to other students. Calculus can be taught using just basic algebra, and most students will benefit from already understanding calculus, when they are learning trigonometry.

Trig is artificially hard in calculus. When you get to complex numbers it all gets easier. But they don't teach that until later. One thing I learned at Berkeley was that there are two kinds of problems: linear problems and problems you can't solve. The trick ( EE 120 Linear Systems ) was always how to transform a complicated problem into a linear problem. Yeah we used complex numbers as part of the trick to get to l…

> Trig is artificially hard in calculus. When you get to complex numbers it all gets easier. But they don't teach that until later.

And if Clifford algebras were introduced, it would get even easier. But they typically don't even teach that.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#74

Post by eiji »

johan_larson wrote: post #73

When engineers and scientists have to get some real work done with DEs, what do they do? Is it all solved numerically these days?

Ansys, Abaqus CAE, Dyna, Optistruct, FEKO, AcuSolve, NASTRAN, ... The list goes on and on. Each bigger DE or partial DE has a huge ecosystem of numerical software around it to approximate solutions for real world applications.

High and low electro magnetics (Maxwell), Navir-Stokes, ...

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#75

Post by thanatropism »

eiji wrote: post #74
johan_larson wrote: post #73

When engineers and scientists have to get some real work done with DEs, what do they do? Is it all solved numerically these days?

Ansys, Abaqus CAE, Dyna, Optistruct, FEKO, AcuSolve, NASTRAN, ... The list goes on and on. Each bigger DE or partial DE has a huge ecosystem of numerical software around it to approximate solutions for real world applications. High and low electro magnetics (Maxwell), Navir-Stokes, ...

Engineers should be taught about actual numerical algorithms and their qualitative properties. Cases in which there exist explicit symplectic algorithms, etc.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#76

Post by swiley »

stephengillie wrote: post #2

Please teach calculus before teaching trigonometry. There's no prerequisite to learn trig first, and forcing people to learn trig-calc excites many mathophiles but is a major turn off to other students. Calculus can be taught using just basic algebra, and most students will benefit from already understanding calculus, when they are learning trigonometry.

This right here! Calculus (the first semester anyway) is really not hard to understand at all. Things like limits also demonstrate why math is done the way it is which makes it much more enjoyable, but most people get hung up with trig and don't see any of this.

Part of the reason may be that trig gets used heavily for a lot of stuff after calc1.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#77

Post by chillingeffect »

eiji wrote: post #74
johan_larson wrote: post #73

When engineers and scientists have to get some real work done with DEs, what do they do? Is it all solved numerically these days?

Ansys, Abaqus CAE, Dyna, Optistruct, FEKO, AcuSolve, NASTRAN, ... The list goes on and on. Each bigger DE or partial DE has a huge ecosystem of numerical software around it to approximate solutions for real world applications. High and low electro magnetics (Maxwell), Navir-Stokes, ...

What tools are used for parameter estimation? E.g. I made a model that describes my system behavior, then collected a lot of data and next need to find the constants in the model that make the best fit. Thank you.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#78

Post by gh02t »

woopwoop wrote: post #50

I TA'd for an undergrad ODE course for two semesters. I'm a grad student at a public university with very good undergrad engineering students. Nonetheless, with the course I taught, at least, the main problem was that WAY too much was packed into a single semester. I suspect that this is the same everywhere. For example, about a month and a half of the course was devoted to systems of linear equations. These were stu…

> These were students who, a priori, knew basically no linear algebra, being asked to understand an entire linear algebra course, with some extra stuff thrown in (you know, the differential equations), in six weeks.

Yep, Diff Eq 1 (ODEs) and Diff Eq 2 (PDEs) for us as undergrads were exercises in memorization, even as a math major. I didn't really understand them until after I took a (well, several) linear algebra classes, after which it clicked that it was basically all the same theory.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#79

Post by eli_gottlieb »

woopwoop wrote: post #50

I TA'd for an undergrad ODE course for two semesters. I'm a grad student at a public university with very good undergrad engineering students. Nonetheless, with the course I taught, at least, the main problem was that WAY too much was packed into a single semester. I suspect that this is the same everywhere. For example, about a month and a half of the course was devoted to systems of linear equations. These were stu…

You don't make students take Linear Algebra before differential equations? I thought students were supposed to take Lin-Alg alongside their Calculus 2 or Multivariable Calculus course, then have all that in their heads for DiffEqu.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#80

Post by trentmb »

stephengillie wrote: post #28
jacquesm wrote: post #3

Earlier quoted context omitted.

I found trigonometry a lot easer than calculus, which even today usually takes me forever even for what I'm sure are simple problems.

I found sine and associated concepts to be arcane and Byzantine when learning then through algebra and geometry alone. Seeing sine, cosine, etc as merely each other's derivative was astonishing and eye opening. So elegant. It made me love math again.

The relationship between sine and cosine is apparent just by looking at the unit circle.
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