3 ms·
Do you also think the nuance of co- and comtravariant tensors should be understood, and the conventions of raising and lowering indexes, which is where a lot of
by reikonomusha 6y ago
Do you also think the nuance of co- and comtravariant tensors should be understood, and the conventions of raising and lowering indexes, which is where a lot of value of Einstein notation comes from?
- MiroF 6y agoYes, much more preferable to 4 years of talking about "row vectors" and "column vectors." Perhaps not so early though. Introducing the concepts in the context of a euclidean metric seems unmotivated?
- ssivark 6y agoWhile not necessarily in a first course, it is an important difference to appreciate which saves a lot of headache later. Eg: Vanilla gradient descent involves naively adding a covariant and a contravariant quantity, which is generically a recipe for disaster, and nicely motivates second order methods that use the Hessian as the "metric" to raise/lower indices. The picture about Hessian eigenvalues determining the step-size is of course complementary and helps gain an intuition for the meaning of a "metric" and its importance.