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19h
comment Light attenuation through water at an angle
The light reflects and refracts as it enters the water. This makes a significant difference when we consider different angles. However, after the interface is crossed, it goes on as if it was always wet.
20h
comment Differntial Geometry
circular motion at constant speed.
1d
comment Review on Riemannian Geometry
Maybe amazon.com/Lectures-Algebraic-Geometry-Applications-Mathematics/… would be of interest. Another book which gets very far very fast is amazon.com/… which is probably better for a review than a first read of the topic.
1d
comment Review on Riemannian Geometry
Certainly not an algebraic geometry text, but, Kuhnel has a relatively compact treatment of Riemannian geometry in ams.org/publications/authors/books/postpub/stml-16-R That said, as an outsider, what I read about tangent spaces from an algebraic-geometry perspective sounded like some sort of hidden art masked by secret terms... good luck. I hope someone comes along with a better answer than this comment.
2d
comment Difficult Coordinate Geometry and Calculus Question
Can we argue by symmetry this must be the answer?
2d
comment Levi-Civita tensor
I would argue that this "property" is actually the honest definition of the determinant. The expansion by minors due to Laplace is far more complicated even if it is how you first see the determinant defined. Of course, mathematicians who are found of permutations can differ with my opinion, but, I prefer a sum of indices as opposed to a sum over permutations (the other good way to define the determinant)
2d
revised Multiplying two tensors of the Levi-Civita type
added detail.
2d
comment Multiplying two tensors of the Levi-Civita type
@PhilosophicalPhysics I added a sentence to your question. Please edit it if you disagree with what I said. I simply put what I thought was your confusion. I hope your question can get unholded with this edit.
2d
revised Multiplying two tensors of the Levi-Civita type
added comment to make the confusion explicit.
May
23
comment Multiplying two tensors of the Levi-Civita type
The second expression is the same as the first since he is able to shift the indices two times over to obtain the second expression. It follows from the complete antisymmetry of the levi-civita symbol.
May
23
answered Multiplying two tensors of the Levi-Civita type
May
23
comment Multiplying two tensors of the Levi-Civita type
by conservation of indices, you are missing a prime in the first equation's rhs.
May
23
comment About summer course or online course of Linear algebra and real anyasis
It's not quite at the level you want, but, perhaps youtube.com/playlist?list=PLBY4G2o7DhF1GaQaG4haiIGKk9qEYxCZQ will help you. I use a combination of my own notes and Damiano and Little's nice text (available as a Dover)
May
22
reviewed Approve Orthogonal projection matrix
May
22
reviewed Approve Is group $G$ must abelian, when some condition is given by using exact sequence?
May
22
revised Derivative of matrix exponential wrt to each element of Matrix
added 2337 characters in body
May
21
comment Are partial derivatives a special case of the total derivative or just something else entirely?
A partial derivative is a special case of a directional derivative. A directional derivative exists when the function is differentiable. Differentiability in the general sense is far more subtle than the existence of directional derivatives... see posts like math.stackexchange.com/q/503632/36530
May
20
answered Derivative of matrix exponential wrt to each element of Matrix
May
19
comment Can I combine the wave and heat equations?
I think you could use the usual techniques we solve the wave and heat equations to solve this problem. Can you just take a solution off the shelf to solve it? I don't think it fits the standard templates I know.
May
19
comment Continuity and differentiability of the function $x|x|$
the function is once, but, not twice differentiable at zero.