Lie derivative of a vector field (flow and pushforward)
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- Опубликовано: 6 сен 2024
- Part 2: • Video
In this video I show you how you can derive the Lie derivative of a vector field. First, we look at a vector field on a manifold and develop the notion of an integral curve followed by the flow of the vector field. We can then move another vector along that flow, moving it to a different tangent space. To be able to compare tangent spaces we need the induced map, or pushforward. Using the pushforward we are finalle able to relate vectors in different tangent spaces and we can find an expression for the Lie derivative.
The program you see my using is called "Write" and I greatly recommend it. Check out Write at styluslabs.com
The font used in the video is called Inconsolata by Raph Levien. It's a beautiful monospaced font and if you do any programming at all and are looking for a different font style for your editor you might want to check out his website: levien.com/typ...
Helped me sink down a bunch of concepts that I've been reading but wasn't able to connect all together. Really nice how you threaded the story! Thx
This is the best video I've encountered on the Lie derivative. Thanks a lot, I'm sad because I didn't find your channe earlier. Gonna binge watch all these videos to see if I can pass tomorrow's exam.
It just amazes me how a simple idea can be tarted up with notation and made to seem impenetrable. I will never understand the modern ethos in math. Thank god everything I need is from earlier days and most of this noise can be ignored.
Glad you returned with another great video
You are really awesome. Thank u
Really good video.
thank you for this video
Excellent video. One comment, would the computation be simpler if we find a nice coordinate chart in which X=\frac{\partial}{\partial x_1}? Then the flow asscoiated with $X$ is just translation along $x_1$ direction?
I assume the equation just finished by 15:51 implies Einstein's summation, and Y and Y\tilde are of same dimention, is that right?
Yes, that's right
I don't think description at 21:39 is correct.
1. $X^v$ is a vector field that takes $x \in M$, so the expression $X^v(x^\alpha + ...)$ is not well-defined.
So, you really should consider multidimensional taylor expansion. Overall, great video!
Hmm I'm not sure what you mean. The argument of X^v is the point q that is also an element of M. At around 20:00 I essentially derive q^v = x_0^v + \epsilon X^v. This expression has an error orf order \epsilon^2, but since \epsilon is infinitesmial anyway that doesn't matter.
my ma‘am