The Most Important Video in the Series: The Linear Property of the Fundamental PDEs
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- เผยแพร่เมื่อ 15 ก.ย. 2024
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Hi, Pavel here. Someone was very kind to provide closed captions for this and other videos. However, I can't seem to figure out how to determine who made the contribution. Can someone let me know who to determine the author so I can thank them properly?
Thank you in advance!
Pavel
Professor MathTheBeautiful, thank you for a solid explanation of The Linear Property of Fundamental PDEs. This is an error free video/lecture on TH-cam TV.
06:30 Alan Turing once wrote on a postcard to his friend Robin Gandy:
"Science is a differential equation. Religion is a boundary condition." ;)
02:35 So what is the linear algebra analog for the other two (time-dependent) types of equations? :q
My guess is that they're actually ALL of the type `A·x=b` (where `b` might be `0`), but their operators now take time into consideration as well. For example, the wave equation can also be written as:
□² u = 0
where □² is the d'Alembert operator:
□² = (1/c²)·∂²/dt² - ∇²
03:39 `x³ - 3·x²·y` is not harmonic:
(∂/∂x)(x³ - 3·x²·y) = 3·x² - 6·x·y, (∂/∂x)(3·x² - 6·x·y) = 6·x - 6·y
(∂/∂y)(x³ - 3·x²·y) = -3·x², (∂/∂y)(-3·x²) = 0
So when added together they don't produce 0 :q
Perhaps you had in mind this one?: x³ - 3·x·y²
This one is harmonic:
(∂/∂x)(x³ - 3·x·y²) = 3·x² - 3·y², (∂/∂x)(3·x² - 3·y²) = 6·x
(∂/∂y)(x³ - 3·x·y²) = -6·x·y, (∂/∂y)(-6·x·y) = -6·x
which when added together produce 0.
Fun fact:
(x+y·i)² = x² + 2·x·y·i + (y·i)² = x² + 2·x·y·i - y² = (x²-y²) + i·(2·x·y)
The real and imaginary part are the two harmonic functions you mentioned ;> `x²-y²` and `2·x·y`.
(x+y·i)³ = x³ + 3·x²·(y·i) + 3·x·(y·i)² + (y·i)³ =
= x³ + 3·x²·y·i - 3·x·y² - y³·i =
= (x³ - 3·x·y²) + i·(3·x²·y - y³)
This gives another two harmonic functions: `x³ - 3·x·y²` and `3·x²·y - y³.
And you can do the same with higher powers of a complex number `x + y·i` to obtain more of harmonic functions :)
How cool is that? ;>
I'm not use to his notation.... its different from 3blue1brown