Tensor Calculus and Chill

Admittedly, family entertainment is pretty dumb.

me too!

Or is it

It's like farting under the quilt. Secretly she loves it.

'top 10 anime showdowns - watchmojo'

I think they must have made sense to people who tried working with a whole bunch of awkward to write sums, got used to the constant patterns, and then just started writing the patterns. Also, I watched this video and half of the next one until I figured out that he had actually explained near the beginning of this video that these were ways of writing sums of monomials. I thought he had spent an inordinate amount of time explaining rules for things whose meaning he hadn't told us until he started talking about theorums and I realized that he had told us what these things meant.

its actually genius, information density per caracter is so much higher

😂😂😂

- 3rd year astrophysics student

@@ameerbux78666 in which university?

@@llayashree3974 Witwatersrand

Glad you like them!

Glad it was helpful!

Smoothdraw + a bamboo tablet.

Is 6D rank of the tensor here?

Rule 3 says that you can't have the *same* index occur three or more times in a given term. If you have a 5-D tensor for instance, you're allowed to write it as a_ijkmn*b_i (with 5 different indices - i, j, k, m, n). However, you can't write something like a_ijkmn*b_ii, where the i occurs 3 times. Of course, with higher dimensions, you will require more indices to describe your tensor. Hopefully that clarifies things!

I can't think of any: pretty much every physical quantity falls into one of the tensor categories, either scalar, vector or rank-2+ tensor.

Possibly you didnt follow this from 'THE BEGINNIG' or ... >> I see it costs too much in usa -- leonardo

@Faculty of Khan Hmm but if we're considering invariance then isn't only the spacetime interval supposed to be invariant? Then the space and time separations are co-ordinate dependent and thus not tensors.

Yes! For higher-order tensors, like rank-3 for instance, we'd be using another unique index like k (e.g. A_ijk).

The indices in different positions denote different vector types. By most usual conventions, a subscript denotes a component of a row vector, while a superscript denotes a component of a column vector: en.wikipedia.org/wiki/Einstein_notation#Mnemonics Hope that helps!

you could try a student's guide to vectors and tensors by daniel fleisch

Thanks, I'll try it

What do you mean exactly by one index up and one down? Like instead of a_ij*b_j, you use a_ij*b^j?

we always wrote it like a^j_i*b_j if you're summing over j. like a_ij*b_j is one member of a sum, when multiplying a matrix A=(a_ij) and a vector b=(b_j) for example, while a^j_i*b_j is the whole sum

Interesting. I also agree that your convention probably doesn't have any regional ties, but I've seen it used in a bunch of places, so it's likely a book-dependent thing. Here's a couple of examples in Wolfram MathWorld where the index being summed over is in the subscript both times: mathworld.wolfram.com/EinsteinSummation.html I also found a stackexchange question addressing your comment, and it turns out that's it's a book/author-dependent convention: physics.stackexchange.com/questions/262237/einstein-summation-convention-one-as-upper-one-as-lower As another example, the books I'm using right now (Schaum's Outline + Boas Mathematical Physics) don't really use the one upper and one lower convention. But yeah, I agree with you that it might be more conventional to use one index up and the other down, but this is just an introductory video. I feel like when we get deeper into Tensor Algebra and operations involving tensors and vectors, the need to use both upper+lower indices will arise with the introduction of contravariant and covariant tensors. Hope that clarifies things!

Yeah, I don't think the placement of indices matters until you stop using Cartesian tensors.

Upper indices correspond to the components of a tensor that are in the vector space itself while the lower indices usually respond to the indices that arise from dual functionals. If the space has no differences between the two, then they don't matter. Otherwise they done. I have done no degrees at UC Berkeley. lol.

That's what the next video is for!

Faculty of Khan Beautifully presented and explained otherwise. Thank you for your uploads!

Yea, but people can mistake the 'sup' for 'sub' since they sound similar.

@@FacultyofKhan ​ Pronounce it as 'soup' as it is in 'super'

@@gilrutter9481 It's still much easier to tell the difference if one word is one syllable and the other two syllables, since those two vowels are kind of close to each other (very close if you're from Northern England), and the two consonants are both very similar between the two words and are positioned so the two words have similar "shapes" (so to speak). I think one measly syllable per superscript is a small price to pay for having one less thing to be confused about.