But what is a partial differential equation? [video] | Hacker News Reader

But what is a partial differential equation? [video]

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Or as I used to call them, Particularly Difficult Equations.

You see, the trick to a PDE is to use the LOFL transform. The LOFL transform (Leap Of Fuggin Logic) moves your PDE into easy space. Easy space has the important property that all problems are trivial and or linear. Once you've solved you're PDE in easy space you just apply the inverse LOFL and you're done.

How do you take a LOFL transform you ask? That's left as an exercise for the reader.

Better yet - I'll show you how to perform the LOFL. I'll show you with the most trivial example possible. It will have no applicability to any other problem you encounter and may in fact be the only problem solveable by the LOFL.

Integral transformations are the "rest of the fucking owl"[1] of the STEM world.

[1] - https://i.kym-cdn.com/photos/images/original/000/572/078/d6d...

Interesting that you mention this as this is a topic I learned today in Numerical Linear Algebra: Krylov subspace methods.

Wish I had access to 3B1B when I was just starting to learn this stuff. His expository style and graphics are such great sources of intuition.

But, a differetial equation is not an equation, right? It’s a definition. Its solution is an equation. Do I understand this correctly?

No, it's an equation. Equation is a statement that says that thing A is equal with thing B. The point is that thing A and thing B are often quite dissimilar in form. But since they are the same thing, and because math is deterministic, if you manipulate the both sides the same way (For example, multiply both sides with the same amount), it stays equal. (Plus sometimes you are allowed to manipulate them in different ways, because the axiomatic system you're in says that's allowed for some cases. Example: you are allowed to rewrite (AB)C as A(BC) if you know that the object you are manipulating is associative.) "Solving" an equation means manipulating it in a way that the other side becomes trivial in form. You maybe can't do that for many differential equations, but that doesn't make them less of equations.

> Equation is a statement that says that thing A is equal with thing B.

But this describes an equality. I imagine an equation to be a proportionality written with standard units. Aren't they different? I mean an equality, an identity, a definition and an equation are different, even though most of the time they are indicated with the equality sign.

I believe identities and definitions have different symbols by convention, either the equal-delta sign (definition [0]), or the triple-line equals sign (identity [1]).

[0] https://hsm.stackexchange.com/questions/5166/who-first-defin...

[1] https://www.quora.com/What-does-a-triple-equals-sign-mean

Well technically a PDE is an operator between various function spaces.

What "solving" means is a bit arbitrary.

Solving a differential equations typically means taking one of the sides and going down one order.

For example making a new equation 'temperature' = X from 'derivative of temperature' = X

Well I guess they would be equations that are partially differentiated from other equations?

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