Wow, it's been a while. Does anyone actually use this stuff after college???
This is why people suggest that terms like 'software carpentry' more accurately capture what 99.5% of programmers do than 'software engineering'.
Introduction to Differential Equations (2008)
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Re: Introduction to Differential Equations (2008)
#32DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. When I started a gamified music discovery company, I actually ended up using DiffEq to define a scoring algorithm that would produce continuously varying point values based on time series input data. I had to relearn how to do it…
Also, what they usually don't tell you is that the recipe bag only works for toy problems. For real applications you most often need numerical approximations.
Re: Introduction to Differential Equations (2008)
#33Does anyone know websites or resources which explains how Diff Eq is used in Computer Science? I know it's used in a variety of areas in CS, but I really like to see or read well-explained tutorials or articles. e.g. what's finding area got to do with the topics in CS? what does 'area' correspond to?
It's essential in computer graphics or in computational geometry and in simulation science, where I can speak from experience - these are more intersectional topics than pure CS I think but I'd say it's applied CS.
E.g. in finding faster ways to do Ray/surface intersections (if we're talking about actually industrially useful geometry like all kinds of splines and not just triangle meshes), differential geometry is essential - even with triangle meshes you can apply it in normal and curvature estimation. Differential equations and integrating them enter the picture if you want to find the shortest way from one surface point to another along the arbitrarily shaped surface.
With simulations differential equations are everywhere because any physical system as a function of space (and time) is a collection of differential equations that you need to solve.
Earth-movers distance and the Wasserstein metric have recently got attention again, its original relevance was in the Monge-Ampere problem, how to distribute a continuous distribution of 'heaps' of some kind into a distribution of 'sinks' with the least amount of total distance moved. Which is a nonlinear partial differential equation to solve in two dimensions.
We need to apply numerical methods, nonlinear optimization to solve such problems and CS is a part of doing that quickly. Because there are no general closed form solutions for most of the systems of differential equations or there we need algorithms to solve them approximately.
Re: Introduction to Differential Equations (2008)
#34DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. When I started a gamified music discovery company, I actually ended up using DiffEq to define a scoring algorithm that would produce continuously varying point values based on time series input data. I had to relearn how to do it…
Also, what they usually don't tell you is that the recipe bag only works for toy problems. For real applications you most often need numerical approximations.
One of my professor used to say that you can do derivatives of every functions, it's mechanical, but you will only be able to find exact solution of an infinitesimal fraction of diffeq, and thinking about that, the same apply with integrals.
Re: Introduction to Differential Equations (2008)
#35Earlier quoted context omitted.
Yeah, it's relatively dry material which is hard to grasp without the context of why we need it and how it can be applied in the real world. While the intro is good, it still has the same problem - just shows (in a good way) "some math".
I think the calculus of variations might be a better approach to introducing ODEs in first year. You can show that by generalizing calculus so the values are functions rather than real numbers, then trying to find a max/min using the functional version of dy/dx = 0, you end up with an ODE (viz. the Euler-Lagrange equation). This also motivates Lagrange multipliers which are usually taught around the same time as ODEs…
You don't need a Lagrangian to invent mechanical ODEs. You could talk about mixing tanks, objects under complicated forces, and so on with a lot less background information.
Re: Introduction to Differential Equations (2008)
#36I think the schools spend far too much time on symbolic differentiation and integration. This limits the exercises to the kinds of toy problem that yield to those methods. Kids get sidetracked on solving anti-differentiation puzzles, while the fundamentals are relegated to those (largely useless) puzzles. After 20 years of engineering--in almost every case--numerical methods have been the only way forward. In hindsig…
While numerical methods are absolutely critical in practice, analytic methods like you learn in what people call calculus and diff-eq are _absolutely_ essential to understanding the physical world.
You can't actually _understand_ numerical methods without a fairly deep grounding in analytical methods.
The real problem is here is a lack of context. Engineering and most science curriculums take a "short-cut" through mathematical education. They try to teach just enough math to get through the major coursework. As a result you end up with students who feel it's all just one big memorization trick .
Re: Introduction to Differential Equations (2008)
#37I think the schools spend far too much time on symbolic differentiation and integration. This limits the exercises to the kinds of toy problem that yield to those methods. Kids get sidetracked on solving anti-differentiation puzzles, while the fundamentals are relegated to those (largely useless) puzzles. After 20 years of engineering--in almost every case--numerical methods have been the only way forward. In hindsig…
While numerical methods are absolutely critical in practice, analytic methods like you learn in what people call calculus and diff-eq are _absolutely_ essential to understanding the physical world. You can't actually _understand_ numerical methods without a fairly deep grounding in analytical methods. The real problem is here is a lack of context. Engineering and most science curriculums take a "short-cut" through ma…
> analytic methods ... are _absolutely_ essential to understanding the physical world
How so? My experience has been that the "physical world" is where the symbolic approach completely breaks down.
> Engineering and most science curriculums take a "short-cut" through mathematical education.
Only people taking more math than scientists and engineers would be mathematicians. A year-and-a-half course to cover the limit, tangent-at-a-point, functions of tangent-at-a-point, area-under-the-curve, and generalizing all of that to higher dimensions doesn't seem like much of a "short-cut" if you ask me.
Re: Introduction to Differential Equations (2008)
#38Can anyone compare this vs other resources to learn differential equations? I want to learn math roughly to the level of an undergrad engineering student, so I've looked at some Advanced Engineering Mathematics books (one by Zill, another by Kreyszig), both have mostly good reviews, and to be honest, Paul's Notes seem almost a level above in clarity and understandability. For example, compare the explanation of integ…
> compare the explanation of integrating factor and exact equations
those are fringe subjects, completely irrelevant for the modern usage of differential equations. They are useful only in computer algebra when you want to implement differential galois theory. In practice you want to understand the overall behavior of your system (qualitative theory) or compute particular solutions numerically (using numerical methods, which are more precise than evaluating the expression of the exact solution).
You'd do much better with a qualitative book about differential equations (e.g., Arnold), about numerical analysis, or about dynamical systems (e.g. Strogatz).
Re: Introduction to Differential Equations (2008)
#39DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. When I started a gamified music discovery company, I actually ended up using DiffEq to define a scoring algorithm that would produce continuously varying point values based on time series input data. I had to relearn how to do it…
"Ten lessons I wish I had learned before I started teaching differential equations" is relevant here. I feel that DiffEq was the most useless undergraduate course that I took for my comp sci degree. They really didn't spend enough time going into the fundamental concepts so that I am not even sure I could recognize a differential equation if it were staring me in the face at this point... much less any of the tricks…
Sounds like this course, which is very well done.
https://www.thegreatcourses.com/courses/mastering-differenti...
Re: Introduction to Differential Equations (2008)
#40DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. When I started a gamified music discovery company, I actually ended up using DiffEq to define a scoring algorithm that would produce continuously varying point values based on time series input data. I had to relearn how to do it…
The algorithms you would learn for graphs etc are also just math and equally abstract. What makes the difference in how real you treat the two concepts that are both abstract but generally applicable?