This a direct nice basic review of directional derivatives. It's pretty interesting revisiting these old videos and seeing the contrast between them and now. It seems like you have better lighting and edit out some parts nowadays. I still enjoyed the video!
Conjecture: Let f: R^n -> R^m and g: R^m -> R^k. Let h be their composition, h: R^n -> R^k. If P(f) and P(g) then P(h), where P is each of {continuous, differentiable, integrable, analytic}. Here's how to compute the (anti-)derivative or taylor series or whatever: [...]. If true, this suggests a different proof architecture: (1) prove my conjecture. (2) prove that the change-of-basis function in the input space is nice (diff'able, continuous, analytic, whatever). Instead of proving that the combination of two specific steps preserves membership of some class, you get a lot more power if you show that the class is closed under a very generic kind of combination and then show that all the combinators you want are members of that class. For example, most sequence/function classes are closed under linear combination, product, reciprocal, composition, inverse-function, etc., and this gives a lot of power. Oh well, maybe this particular theorem is one step in showing that this more generic composition framework works.
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with your video, now there is no question in my mind about gradient and directional derivatives. Thank you
This a direct nice basic review of directional derivatives.
It's pretty interesting revisiting these old videos and seeing the contrast between them and now.
It seems like you have better lighting and edit out some parts nowadays.
I still enjoyed the video!
Thanks for the transparant proof .
Conjecture:
Let f: R^n -> R^m and g: R^m -> R^k. Let h be their composition, h: R^n -> R^k.
If P(f) and P(g) then P(h), where P is each of {continuous, differentiable, integrable, analytic}. Here's how to compute the (anti-)derivative or taylor series or whatever: [...].
If true, this suggests a different proof architecture:
(1) prove my conjecture.
(2) prove that the change-of-basis function in the input space is nice (diff'able, continuous, analytic, whatever).
Instead of proving that the combination of two specific steps preserves membership of some class, you get a lot more power if you show that the class is closed under a very generic kind of combination and then show that all the combinators you want are members of that class.
For example, most sequence/function classes are closed under linear combination, product, reciprocal, composition, inverse-function, etc., and this gives a lot of power.
Oh well, maybe this particular theorem is one step in showing that this more generic composition framework works.
You could have talked about the geometrical significance of the definition of the directional derivative.
3:58 Won't u and v depend on x and y respectively too?