I explained the tensor contraction with details by showing some examples. This is for the general relativity course. I hope this helps you study GR easier !
[Correction] At 3:17 , to be more exact, I should have shown the given vector as a column(vertical) array, since it's a contravariant vector. When it's a single line, I just don't care much about writing it horizontal or vertical. Sorry if this confused you ! If this video helped you, pleaseeeee subscribe to my channel. It really motivates me ! :)
Contravariant is dual to covariant -- vectors or dual spaces. "Always two there are" -- Yoda. Domains (vectors) are dual to co-domains (co-vectors, forms) -- Group theory. Injective is dual to surjective synthesizes bijective or isomorphism (duality).
Heya, I see that sadly you haven't posted anything in the last 6 months, but I hope you see this comment. This was a really good and concise explanation, and it really helped me out, so thanks a lot! The channel has quite a bit of useful videos like this. I know all this must have taken a lot of effort to make, so just wanna let you know it wasn't for nothing!
QUESTION PLEASE: At 3:17 - Can you please explain 1) You multiply the metric tensor/matrix by a column version of Vu, but Vu is not a column, it is a row. How can you just change Vu from a row to a column? and 2). Multiplying the metric tensor/matrix by a column version of Vu should result in another column vector (just try it with MATLAB or Mathematica to see) but your result is a row vector. How can your result be a row vector? I am really struggling with this so if you can explain I would greatly appreciate it.
You are right. To be more exact, I should have shown it as a column from the beginning. Sorry. I tend to not care much about writing it as a column or a row when it's just a single line. You know what I mean? Defining column and row becomes crucial from matrix level. But sorry about the confusion again.
@@ReumiChannel Thank you - and no need to be sorry. Your channel is fantastic and has helped me immensely. I have seen a few of your videos on this subject - some of them multiple times!
Isn't this process the raising and lowering of indices? Contraction is summing over pairs of like upper and lower indices and reducing the rank by 2 for each pair.
@3:34, you have broken conventions. You first show a contravariant tensor as a row vector. Then in the next statement, you show a covariant tensor as a row vector. By convention, contravariant is depicted as column vector and covariant depicted row vector. Also, you said a Tensor is a matrix. This is not correct. There are many instances when a Tensor cannot be demonstrated as a matrix. And the reverse is also true. Not all matrix are tensors. Not all matrices obey a well defined transformation rule. Tensors are representations of a physical object which is real independent of it's bases or representational model. Matrices are just mathematical objects. It is the fact that they are real, that certain transformation rules must be obeyed.
Yes. But we don't need to strictly follow "covariant=row contravariant=column" convention, as it no longer becomes useful when dealing with higher rank tensors. Not significantly important when learning about tensors. At least I didn't have any problems when studying Physics. I may have unintendedly called it a matrix, but I don't see that as a big problem. I was in fact showing the math. As you said, matrix is a mathematical object. I tend to call things in a more friendly and comprehensible way for the new learners. When learning for the first time, you cannot learn everything quickly with exact definitions. It would be the best if we could always do that, but my belief is, it doesn't quite work that way in reality. If you watch my other videos, I try my best to mention "this is not exactly true, but for now you can understand the definition this way". It's always better to focus on just one thing. Thanks for the corrections. You are not wrong, and I do hope some people read your comment and take more exact information. I don't want to add anymore details to this video though. That is too much information in my opinion. The most important thing is, did you get to learn something useful from this video ? :) i hope you did !
in the future, I would recommend not using symbols that sound the same -- for example, you used mu and nu for the indices, making it really hard to follow.
It normally takes 2 weeks for me to make one video. Also, I don't know if it is urgent for you, but I don't have time at the moment 😢 Just remember these for Noether theorem : If the equation of motion is time-invariant : energy is conserved. If it's translational-invariant : linear momentum is conserved. If it's rotational-invariant : angular momentum is conserved. Once you identify that, you define a Killing vector.
if a student is nill and have low iq but he is willing to learn high math what should he do? which channels to follow ? can you please guide me how to learn finite element method
