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Poisson's Equation

mattferraro.dev

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Re: Poisson's Equation

#141

Post by beerandt »

wombatpm wrote: post #131
H8crilA wrote: post #127

Earlier quoted context omitted.

What's a "test curve"? How does it make 23/100 be an A grade, is it some kind of CDF-based transformation that makes certain % of students pass the test?

Unlike in high school where the grade percentages are pre determined. ie a 90% is the cutoff for an A and it is expected that students will score that or higher, College professors see that as making the test too easy with the potential of having too many students get A's. Instead they will make the test longer or harder such that the best grade is 60/100 and the average grade is around 40. The will then scale things…

Also, the term "curve" is literally a statistical bell curve that teachers would fit the students' grade distribution to.

Even if teachers use a different method, the term has stuck as a synonym for "grade adjustment".

I only had a few actually do this, usually it's just a fixed amount bump given to everyone's grade.

Only 2 or 3 times did I have a teacher curve grades down.

Re: Poisson's Equation

#142

Post by powderpig »

I too hate the coloured plots but I can tell you as a Stress & Structures engineer who works with ANSYS regularly, it does help when assessing material limits and strain energies. This is especially true when you're working with models where temperature is time dependent.

A great article, one to bookmark for sure.

Re: Poisson's Equation

#143

Post by lesquivemeau »

nickponline wrote: post #138
mixedmath wrote: post #44

I think this is a beautiful article. There's code, there's math. There are many plotted examples using a variety of different plotting techniques (in total, representing a 2d array as data, or in black/white, or with a terrible 'jet' colormap, or as a 3d terrain. I very much appreciate this sort of post.

Maybe you'll like these: https://nickp.svbtle.com/ (shameless plug)

Thanks for this

Re: Poisson's Equation

#144

Post by leephillips »

jozvolskyef wrote: post #133
SavantIdiot wrote: post #56

I could have used this 32(!) years ago when I was struggling in college. (This and 3b1b.) It amazes me just how many key topics were so inaccessible to the majority of the class at engineering school. I base this on observations from group study sessions and the hyper-aggressive test curves. I knew lots of people who never got the hang of div/grad/curl, or a Jacobians, or eignenvectors, or Z-transforms... These are k…

The generation growing up today is the first that can broadly learn mathematics from mathematicians rather than from teachers, and we should encourage them to do so. When I was 12-16, I was interested in physics enough that I would study it in my free time just for fun, solving problems from the Physics Olympiad. I could solve lots of the problems with just intuition, until I stumbled upon the sliding chain problem.…

To be fair, sliding chain problems are difficult.

Re: Poisson's Equation

#145

Post by gpsx »

codethief wrote: post #74
jbay808 wrote: post #65

Earlier quoted context omitted.

Isn't Poisson's equation basically describing a minimal surface for small z? I'm not saying the badminton racket follows exactly a (discrete) 2D Poisson equation. But it's certainly related enough to be more than a surface similarly. The cords are under high tension, which means that any curvature along x (that is, dz^2/dx^2) will result in a net z-axis tension force unless balanced by an oppositely curved cord runni…

I didn't say you're completely wrong, either. :) (In fact, I stressed that both equations are of elliptic type, so solutions might look similar.) One can make a similar heuristic argument, though, as to why the surface you observed should follow the minimal-surface equation. EDIT, as you've updated your comment: > Isn't Poisson's equation basically describing a minimal surface for small z? Small changes in z, I think…

If you want to get technical, this isn't a continuous surface but weave of strings. And something that makes this more complex is the location of the string intersections has a static/dynamic friction component that is difficult to model analytically. But I think we accept your approximation if you are looking for numeric results.

Re: Poisson's Equation

#146

Post by jozvolskyef »

leephillips wrote: post #144
jozvolskyef wrote: post #133

Earlier quoted context omitted.

The generation growing up today is the first that can broadly learn mathematics from mathematicians rather than from teachers, and we should encourage them to do so. When I was 12-16, I was interested in physics enough that I would study it in my free time just for fun, solving problems from the Physics Olympiad. I could solve lots of the problems with just intuition, until I stumbled upon the sliding chain problem.…

To be fair, sliding chain problems are difficult.

Thank you for the encouragement. I'm still hoping to solve it one day. No spoilers, please :)

Re: Poisson's Equation

#147

Post by wheelinsupial »

SubiculumCode wrote: post #137
wheelinsupial wrote: post #107

Earlier quoted context omitted.

Is the second section “when would I use this” sufficient to provide a “why you should care statement”? Or is there a more common way to call this out in technical writing? Perhaps an “intended audience” or “necessary prerequisites”?

I mean there is nothing "wrong" with starting at the 2nd paragraph (and that was was just for the more limited Laplace version of Poisson's), its just not generally the most effective writing style. Yes, there is some subjectivity here, but writing is hard, and it is only through criticism that we learn to be better technical writers. The first paragraph of the conclusion would have better served as the Introductory…

> it is only through criticism that we learn to be better technical writers.

Hence the line of questioning. Thanks for the response!

Re: Poisson's Equation

#148

Post by vlmutolo »

just_temp wrote: post #118
vlmutolo wrote: post #78

> It is customary when simulating heat flow to use a wacky color palette where red is hot and blue is cold, with all kinds of intermediate colors in between In an otherwise excellent article, this is the only issue I could find. We really need to stop using/recommending/normalizing rainbow color maps (i.e. jet). They actively confuse readers by creating visual artifacts that aren't actually in the data. This article…

Can not agree with this more, if people want to plot something that is linear please use a perceptually linear colormap! Just a one second glance at the Mona Lisa in rainbow/Jet is enough to make you gouge your eyes out. https://peterjamesthomas.com/2017/09/15/hurricanes-and-data-... For a more technical description the information behind the newer matplotlib defaults, particularly the scipy talk, is great. https://b…

Thanks for the links! I especially like the Mona Lisa example. I'll probably steal that the next time this topic comes up.

Re: Poisson's Equation

#149

Post by woopwoop »

Why is the laplacian so ubiquitous? Well, locally any reasonable PDE is well approximated by a linear one. And linear PDEs are of the form Lu = f for a linear differentiable operator L. But why does the particular case of L = laplacian show up so often in physics? Galilean invariance says that the laws of physics should be invariant under translation and rotation. Checking this against Lu = f, you can verify that one requires that L commute with translations and rotations. That is L(u_h) = (Lu)_h, where u_h(x) = u(x+h), and similarly for rotations. What you can show (pretty easily on the Fourier side) is that every linear differential operator with these properties is a polynomial in the Laplacian.
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