Visualizing the Derivative of a Surface Parametrization

The partial derivatives of a function fit conveniently in a matrix. But what does this matrix represent?

[jonathan.gasser]

This is top notch, if you ever make it back on youtube I'd be glad as hell

[orcodrilo]

Liked it!, I like how you 'animate' the shapes, it brings the notions across nicely!

[evenaicantfigurethisout]

why did you stop? i was hoping for more on topology.

[henriquenunes]

Very very good. Thank you so much and keep up the good work

[chickn]

I believe maybe you should mention that the vector that you apply the derivative matrix to represents derivatives of the parameters s and t with respect to time. In other words the three by two matrix transforms the velocity vector of the square to the velocity vector on the torus. Took me a few minutes to figure out.

[marsupialtube]

Hi, cool videos.

Did you draw the torus by hand? How do you do it accurately?

[elainejeffery]

Wow, cool! What other shapes can you do?

[RamuSingh-mqmy]

Good afternoon please tell me best book of calculus in higher mathematics thanks