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Poisson's Equation

mattferraro.dev

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Re: Poisson's Equation

#131

Post by wombatpm »

H8crilA wrote: post #127
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…

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 such that the top 10% get A, 20% B etc.

Sometime the prof screws up and the test is too hard where the aveage grade is 30. Sometimes some over-achieving git 'busts the curve' and scores a 95 and the next highest grade is a 60 - making it hard for the prof to justify giving A's to the top 10%

Re: Poisson's Equation

#132

Post by SavantIdiot »

MontyCarloHall wrote: post #99
SavantIdiot wrote: post #96

Earlier quoted context omitted.

That's not true at all. When someone drops 200 resumes on your desk and says, "Find 10 candidates to interview by tomorrow", you need some initial sort criteria. The truth, my criteria were: #1: university, #2: GPA, #3: keywords. Sure, I was bitten a few times (I hired an MIT master's student who was utterly helpless), but over the course of years doing this, some patterns emerge, and high-GPA absolutely correlates w…

I agree that GPA is useful as a binary indicator—if it’s abominably low, like a 2.0 in your example, it’s a red flag. But a 4.0 is no more predictive of being a good hire than, say, a 3.3. It’s also very school-dependent. Some places inflate grades a lot more than others. In my own anecdotal experience, I omitted my GPA entirely from my CV when looking for jobs straight out of college and got interviews at every sing…

> But a 4.0 is no more predictive of being a good hire than, say, a 3.3

Ironically I graduated with a 3.4, and I did feel guilty for not passing on resumes with GPAs as low as mine. But as I said, when I had many other tasks to do for work, and then had to stop them all to sort resumes (we all took turns), it was hard to justify excursions when there were so many 4.0s. It is a sad truth that new college grads almost all look the same on paper...

Re: Poisson's Equation

#133

Post by jozvolskyef »

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. I spent about a week pondering over it with no results. I approached my math teacher. He admitted defeat and referred me to the physics teacher. If I remember correctly, the physics teacher avoided admitting he didn't know how to solve it and didn't give me any pointers. That was the last Physics Olympiad problem I tried to solve, after dreaming of attending the competition for years (I should mention that at that time my primary interest already were computers; physics was just a hobby). I wonder whether this could still be a problem today when anything can be found on YouTube. There's a lot of noise, too. How long does it take the average curious kid to find the signal that is 3b1b, MIT OCW, etc?

Re: Poisson's Equation

#134

Post by pbhjpbhj »

MontyCarloHall wrote: post #29
rawtxapp wrote: post #13

Earlier quoted context omitted.

Just to point out, it seems like the author themselves submitted the story to HN. Still I agree with your reasoning, but I think slight clickbaity titles do get clicks which is why we keep seeing them.

Yup. This wouldn’t get nearly the same amount of attention if the title were “Poisson’s equation is the most powerful tool in your toolbox for finding steady-state solutions to the heat equation with arbitrarily placed sources ”

Yes, I would love it if HN had optional descriptive titles as subtitles to the main story titles (which would then always be the title given by the author at the time of submission).

Re: Poisson's Equation

#135

Post by yccs27 »

mferraro89 wrote: post #72
wildmanx wrote: post #11

Earlier quoted context omitted.

Also, what's confusing is that algebra usually uses matrices to describe linear functions from n-dimensional to m-dimensional vector spaces. Matrix has n rows, m columns, you give it an n-dim vector and after matrix multiplication you get back an m-dim vector. The author uses a matrix quite differently. You give it two integer coordinates i and j and it gives you the value at position (i, j) back. That's a valid use,…

Thanks for calling this out, I thought it might cause confusion. Matrices are super weird objects because they don't fit nicely into the {scalar, vector, function, operator} classes that maybe we're used to. A matrix is a function in that it can take in a vector and map it to a new vector. It is also an operator in that it can take in some other matrix (a function!) and give you a transformed matrix (a new function).…

Just writing down the connections here for myself and maybe others:

A matrix represents a linear function taking a vector and returning a vector, written as

w = M v

Matrix multiplication corresponds to function composition.

Vectors can be indexed, and we can view them as a function i -> v[i] defined on the indexing set. We can also define basis vectors b_i, such that b_i[j] is 1 at index j=i and 0 otherwise. Any vector can be written as a weighted sum of basis vectors, with the vector components as coefficients:

v = Σ_i v[i] b_i

where Σ_i represents summation over the index i.

Matrices can be indexed with two indices, and this is closely related to vector indexing: For a matrix M, we have

M[i, j] = (M b_j)[i]

Each column of the matrix represents its output for a certain basis vector as input. By writing a vector as a sum of basis vectors and using linearity, we get the well-known matrix-vector multiplication formula:

(M v)[i] = Σ_j M[i, j] v[j]

Re: Poisson's Equation

#136

Post by texaslonghorn3 »

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.…

It is certainly much easier. But if that kind of self study culture doesn't already exist in one's own school, either encouraged by teachers or classmates, it may take a much bigger leap to discover that material.

Re: Poisson's Equation

#137

Post by SubiculumCode »

wheelinsupial wrote: post #107
SubiculumCode wrote: post #77

Earlier quoted context omitted.

It IS a good article, but the audience of the article is fairly specific yet left unspecified. As a general principal in technical writing it is advisable to start with a "why you should care" statement because this naturally informs the reader about the context.

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 remark: "Poisson's equation comes up in many domains. Once you know how to recognize it and solve it, you will be capable of simulating a very wide range of physical phenomena." That is a great sentence. I'm interested now, and I know its context.

Re: Poisson's Equation

#138

Post by nickponline »

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)

Re: Poisson's Equation

#139

Post by [deleted] »

wildmanx wrote: post #9
zitterbewegung wrote: post #3

Sorry but can there be more context to why it is a powerful tool?

This is a typical issue with HN posts. Some poor soul wrote a somewhat competent and maybe even lengthy blog post / article about something they really care about and are knowledgeable about. It may be directed at a specific audience, or maybe just screaming into the void to record down some insight they had for themselves to read again later, or similar. And they use a more-than-necessary general title like "the bes…

[deleted]

Re: Poisson's Equation

#140

Post by SubiculumCode »

mferraro89 wrote: post #100
cornel_io wrote: post #83

> This oval (called a separatrix) has the special property that no wind flows through it at all. It acts just like a solid surface. We have already assembled a reasonably accurate simulation of how air flows around an ellipse in some confined space like a wind tunnel! My understanding of airflow simulation (undergrad-level at best) is that the correct boundary condition is almost without exception the no-slip one: ai…

You are correct! The simulation as written does not handle the no slip condition. The simulated "air" is inviscid and irrotational. You would need to tackle a few more things in order to have a really accurate simulation.

Thanks for the awesome write-up.writer's note. The first paragraph of your conclusion would have served better as your introductory paragraph.
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