Here are two relevant lectures on the algebraic and differential geometry behind tensors. See The Power of Thought, Wildberger on Chromogeometry, All About e and Tensor Calculus.
First the history, going back to Huygens and the use of involutes and evolutes and how these ideas led to Gauss' characterization of the curvature of a surface.
The next extends these basic ideas to ideas of Tangent Spaces
And the second half of the last lecture connects back to discrete spaces and all the way back to Euclid and the idea of solid angle.
Now you be able to see how dual spaces of polynomial approximations give you the objects in Tensor Calculus
These lectures give a clear idea of the connection between parametric splines and polynomial approximations
See The Power of Thought, for more on splines and numerical methods.
First the history, going back to Huygens and the use of involutes and evolutes and how these ideas led to Gauss' characterization of the curvature of a surface.
The next extends these basic ideas to ideas of Tangent Spaces
And the second half of the last lecture connects back to discrete spaces and all the way back to Euclid and the idea of solid angle.
Now you be able to see how dual spaces of polynomial approximations give you the objects in Tensor Calculus
These lectures give a clear idea of the connection between parametric splines and polynomial approximations
See The Power of Thought, for more on splines and numerical methods.
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