Tensor space theory I: over a field - Lec 08 - Frederic P Schuller

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Recommends user149705 · 2016-01 · Mathematics Stack Exchange · on “Book recommendation for rigorous multilinear algebra , tensor analysis, manifolds.”

I am looking for recommendation on books about multilinear algebra, tensor analysis, manifolds theory, basically everything to be able to understand basic concepts of general relativity. I am currently following two online courses by Prof Shuller on youtube ( [LINK] ). One of the course is very complete, it starts from propositional logic all the way to differential geometry. The course is really packed cause obviously there is a lot to cover. Prof Shuller make some really good point throughout the course, and at one point he explain that taking the determinant of a matrix is not always defined (i.e, it is defined if the matrix represent an endomorphism, but not define if it represent a bili

Recommends Antoni Parellada · 2017-02 · Mathematics Stack Exchange · on “Understanding the definition of tensors as multilinear maps”

The question arises from the definition of the space of $(p,q)$ tensors as the set of multilinear maps from the Cartesian product of elements of a vector space and its dual onto the field, equipped with addition and s-multiplication rules, given in this series at this point in time [this video] as follows: A $(p,q)$ tensor , $T$ is a MULTILINEAR MAP that takes $p$ copies of $V^*$ and $q$ copies of $V$ and maps multilinearly (linear in each entry) to $k:$ $$T: \underset{p}{\underbrace{V^*\times \cdots \times V^*}}\times \underset{q}{\underbrace{V\times\times \cdots V\times V}} \overset{\sim}\rightarrow K\tag 1$$ The $(p,q)$ TENSOR SPACE is defined as a set: $$\begin{align}T^p_q\,V &= \underset{p}{

Recommends Darksi · 2023-01 · Mathematics Stack Exchange · on “Components of an $(k,l)$-tensor”

I'm watching this lecture [this video] by Frederic Schuller on YouTube trying to understand the structure of the vector space $T^k_l(V)=\underbrace{V^* \otimes \dots \otimes V^*}_{l} \otimes \underbrace{V \otimes \dots \otimes V}_{k}$ and I have some difficulties with the components. If $(e_1, \dots ,e_n)$ is a basis for $V$ and $(\epsilon^1, \dots, \epsilon^n)$ is the dual basis for $V^*$ , then the basis for $T^k_l(V)$ is of the form $$e_{j_1} \otimes \dots \otimes e_{j_l} \otimes \epsilon^{i_1} \otimes \dots \otimes \epsilon^{i_k}.$$ Now any tensor $F \in T^k_l(V)$ can be written in the form $$F=F^{j_1 \dots j_l}_{i_1 \dots i_k}e_{j_1} \otimes \dots \otimes e_{j_l} \otimes \epsilon^{i_1} \otim

Recommends weirdmath · 2023-01 · Mathematics Stack Exchange · on “On the Meaning of Tensor Components (And Whether a Certain Computation Is Correct)”

I've been recently following Lecture 8 of Frederic Schuller's Geometrical Anatomy of Theoretical Physics Lecture Series [this video] . He defines the components of a type $(p, q)$ tensor as $$T^{a_1 \dots a_p}_{b_1 \dots b_q} \equiv T(\epsilon^{a_1}, \dots ,\epsilon^{a_p}, e_{b_1}, \dots, e_{b_q}),$$ and then claims that it's easy to show that $$T = T^{a_1 \dots a_p}_{b_1 \dots b_q}(e_{a_1} \otimes \dots \otimes e_{a_p} \otimes \epsilon^{b_1} \otimes \dots \otimes \epsilon^{b_q})$$ where $\{e_i\}$ are the basis vectors on V and $\{\epsilon^j \}$ is the dual basis on $V^{*}$ . When I attempt to show that I get, \begin{align*} &T^{a_1 \dots a_p}_{b_1 \dots b_q}(e_{a_1} \otimes \dots \otimes e_{a_p}

Recommends Ted Black · 2024-11 · Mathematics Stack Exchange · on “Notation for Raising and Lowering indices does not agree with matrix multiplication?”

You can think of a mixed tensor like ${X^j}_i$ as a matrix; it is a very bad idea to think of a second order covariant tensor like $F_{\alpha\beta}$ or a second order contravariant tensor like the metric tensor $\eta^{\mu\alpha}$ as a matrix. Under a change of basis a matrix $X$ transforms as $L^{-1}XL$ where $L$ is the matrix that relates the two bases. Under a change of basis a second order covariant tensor $F$ transforms as $L^\top F L $. Have a look at this video [this video] at 1:57:00.

Cites Taylor Rendon · 2021-02 · Mathematics Stack Exchange · on “Possible error in Wikipedia article for $(p,q)$ tensor space notation”

I am currently studying tensor space theory (over a field) and was reading through the generic Wikipedia article for tensors [LINK] when I cam across the following: A type $(p,q)$ tensor $T$ is a multilinear map, $T: \underbrace{V^*\times...\times V^*}_\text{p-copies}\times\underbrace{V\times...\times V}_\text{q-copies} \to K$ where $K$ is the underlying field. Where the author(s) then goes onto define the notation for the tensor product as $\underbrace{V\otimes...\otimes V}_\text{p-copies} \otimes \underbrace {V^* \otimes...\otimes V^*}_\text{q-copies}$ My question: why did they flip the usage of the $p$ 's and $q$ 's? Namely, they first called copies of the dual space $p$ and copies of the