Differential Geometry: Lecture 15 part 1: Shape Operator Defined

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Here we define the shape operator for a surface in R3. Several examples are given, but keep in mind the real computational machinery is given in the next lecture.
  1. James Cook Math
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*Two mathematicians walk into a bar*
(Let's call them Bob and John)

Bob says "Guess what? I finally managed to prove theorem xyz!".
To which his counterpart responds "Really? How did you do it?!"
"Well... It turns out it's trivial:" answers Bob.

Now follows a one and a half hours, three drinks long explanation of the proof.

Once demonstrated, John concurs "You're right, it is trivial."

vn
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NOTE: In this video, U_1, U_2, U_3 are used as the cartesian basis vectors.
This is a head's up for people like me who came to this video - without watching the previous ones in the list. Thanks for the explanations!

nahblue
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I have quite silly question, but I don't get meaning of, for example U_3[cos(theta)], when you say that vector acts on function. Is that a way of writnig a covariant derivative of function cos(theta) in U_3 direction ? I am not familiar with O'Neils book and I guess my teachers use different symbol for that.

lotri
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Thank you very very much for your effort!

KC-kuvm
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Isn't S_p(e_1) = -e_2 and S_p(e_2) = -e_1?

TheMusicDoctor