Basics of Tensor Calculus Differential Geometry Reading Stream Episode 14
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- Опубліковано 19 тра 2024
- You've just wandered into the coolest part of the math streaming ecosystem.
I had filmed reading comments and ranting about philosophy, physics, and history before this but it didn't turn out very well lol.
Last week idk if you noticed/if you've been here before but I typically do 1 hr -1 hr 30 min long videos each uploaded all at once but last week I experimented and split what would've been one long video into multiple shorter ~30ish min videos uploaded throughout the week.
This week, I'm front-loading uploading the reading and doing the fun/casual "just chatting" upload later this week.
Watch my video, then watch these two other videos and skim the following wikipedia article and not understanding tensors will become a thing of the past:
• What's a Tensor?
• Tensors Explained Intu...
en.wikipedia.org/wiki/Tensor
Get the Kreyszig Differential Geometry book I've been reading in these videos: amzn.to/3U6OnbH
📩 Questions/Business inquiries: anthonymakesvideos.info@gmail.com
TIMESTAMPS
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0:26 Starting to read § 31. Basic rules of tensor calculus
0:34 Tensor addition
1:20 Tensor component-wise multiplication by a scalar
2:17 Tensor product/outer product
4:25 Contraction of a tensor
7:47 Tensor inner product
8:47 Problem 31.1 Prove is a^{\alpha\beta} is a symmetric tensor and b_{\gamma\kappa} is a skew-symmetric one, a^{\alpha\beta}b_{\alpha\beta} = 0.
9:03 Problem 31.1 Proof.
9:55 Problem 31.2 Successive coordinate transformations applied to components of tensor.
10:42 Problem 31.2 Solution.
12:27 Starting to read § 32. Vectors in a surface. The contravariant metric tensor.
18:38 Theorem 32.1
22:04 Theorem 32.2
28:19 g_{\alpha\gamma} looks like it says gay
29:09 Lowering superscripts/raising subscripts
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16:01 Contravariant vector -> components calculated from parallel projection
Covariant vector -> components calculated from perpendicular projection
8:47 Problem 31.1 illustrates the *orthogonal nature of symmetric and skew-symmetric tensors* in the context of *tensor inner products.*
-> This orthogonality *simplifies calculations* (result is zero) in Physics, and Engineering, and helps in the analysis of tensor properties.
9:55 Problem 31.2 solidifies the transformation rules of tensor components, showing how *tensorial properties are preserved under coordinate changes.*
@1:43 I struggled to understand the outer product. By definition the Outer product contains both the inner product and determinant (commutator, cross product....). A=[a b c], Z=[x y z] then the outer product is a 3x3. Trace = Inner = ax+by+cz The remaining 6 blocks: just one subtract the other in a special order, it is visually pleasing (that is why I am not doing it for you). Thus 3 block for the inner and 3 commutators, and all 9 blocks are accounted for.
@28:28 Newton high-fives Emmy.....allegedly
Yeah outer product making a bigger data structure and the term "inner product" 's broad usage outside tensors can be confusing. Depends on the problem/situation and we gotta stay cautious out here
Love your videos
Thanks bruv 🤙
Could you help me with these concepts? I have tried to organize these notes. Outer product, inner product, and contraction, here we go:
*Tensor product* and *tensor outer product* are often used interchangeably.
-> Both terms describe the process of taking the product of the components of the original tensors *without any summation over indices.*
-> The term *outer product* is often used to emphasize that the operation produces a higher-rank tensor.
*Tensor contraction* is an operation that involves *summing over a specific pair of indices,* either within a *single tensor* or between *two tensors.*
-> Contraction always *reduces the rank of the tensor or tensors involved by 2,* simplifying the tensor and the resulting expressions.
-> Examples: Ricci tensor, Stress-energy tensor. 🚀
*Tensor inner product* is a specific type of *tensor contraction* that results in a scalar or lower-rank tensors.
-> The inner product of two tensors is a *contraction of their outer product.* 🤔
-> The relationship between the inner product and tensor contraction comes into play when considering the inner product of two tensors of the same rank. 🤔🤔
-> When discussing *tensors of higher rank,* contraction is the preferred term.🤔🤔🤔🤔
-> When referring to operations that result in *scalars* or in contexts where the analogy to *vector dot products* is helpful, inner product is the preferred term.🤔🤔🤔🤔
Terminology issues:
- Traditionally, tensor contraction can involve a single tensor or two tensors, while tensor inner product traditionally involves two vectors. The result of the inner product is typically a scalar.
- While the term inner product is traditionally associated with vectors, the operation on higher-rank tensors is often referred to as a contraction, even though the inner product can be seen as a specific type of contraction where the result is a scalar. 🤔🤔🤔🤔// This contradicts the text above. Should "inner product" be used only where the result is a scalar?
Contractions and inner products are important for defining *tensor invariants,* which remain unchanged under coordinate transformations.💡
-> Tensor invariants are essential in both theoretical and applied physics.💡
*Metric tensors* are not the result of a single tensor operation like the outer product, contraction, or inner product. *However, they can be involved in and defined through these operations.*
-> Example: the inner product of vectors in curved spaces is directly defined using the metric tensor.
This comment and specifically 'Should "inner product" be used only where the result is a scalar?' is featured in the latest episode 16. And that's why they call him @VanDerHaegenTheStampede 😎