# Tagged Questions

Use this tag for questions about specific tensors (curvature tensor, stress tensor), or questions regarding tensor computations as they appear in multivariable calculus and differential/Riemannian geometry (specifically, when it is amenable to be treated as objects with multiple indices that ...

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### Multivariable Calculus with Tensors

I'm looking for a book at the undergraduate level on multivariable calculus (for a 2nd course of multivariable calculus) that introduces and makes use of tensors to describe higher order derivatives ...
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### Divergence theorem for a second order tensor

I want to integrate by part the following integral in cylindrical coordinates $$\int \vec{r} \times (\nabla \cdot \overline{T}) ~d^3\vec{r}$$ where $\overline{T}$ is a second order symmetric tensor ...
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### Tensor contraction

Given that: $T_{i,j}=\lambda\theta\delta_{i,j} + 2\mu E_{i,j}$ Show that: $T_{i,i} = 3\theta \lambda + 2\mu E_{i,i}$ I didn't get the intuition behind tensor contraction, thus i can not solve this ...
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### Matrix Calculus

My school offers Matrix Calculus class next semester. I have never heard about this subject before and got intrigued. After a short chat with professor I found myself unable to get rid of suspicion ...
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### Lorentz transformation of electromagnetic field tensor

I need to calculate: $f^{\mu'\nu'}=L^{\mu}_{\kappa}L^\nu_\lambda f^{\kappa\lambda}$ Where $L^\nu_\lambda$ is the usual Lorentz transformation matrix I thought that I just needed to do some normal ...
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### Relationship between euclidean metric in sphere of radius $r$ and the unit sphere.

I want to show $g_r=r^2g_1$ where $g_1$ is the (Riemannian) metric in the unit sphere induced by its inclusion in $\mathbb{R}^n$ and $g_r$ is the metric in the sphere of radius $r$ also induced by ...
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### Understanding Extension of Scalars in a Vector Space

$\newcommand{\R}{\mathbf R}\newcommand{\C}{\mathbf C}$ Low-Tech Complexification: Let $V$ be a finite dimensional vector space over $\R$. We can forcefully make $W:=V\times V$ into a complex vector ...
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### Levi-Civita tensor in curved space

In the book "Gravitation and cosmology" by Weinberg at the page 99-100. He defines the Levi-Civita tensor as $\epsilon^{0123}=+1$ from which he writes ...
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### Multiplying two tensors of the Levi-Civita type

How to multiply two epsilons with one another? We know ...
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### Covariant metric tensor of a subspace

Suppose $f_1,f_2$ and $f_3$ are vectors in a vector space $V$ with a dot product. Me assume that the vectors are linearly independent. What does it mean to find the covariant metric tensor of ...
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### Find all components of the tensors $T_{ij}=u_iv_j+v_iw_j$ and $S_{ijk}=u_iv_jw_k-v_iu_jw_k+v_iw_ju_k-w_iv_ju_k+w_iu_jv_k-u_iw_jv_k$.

Given three vectors, $\vec u=(u_1,u_2,u_3)$, $\vec v=(v_1,v_2,v_3)$ and $\vec w=(w_1,w_2,w_3)$. Find all components of the tensors $T_{ij}=u_iv_j+v_iw_j$ and ...
1answer
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### Tensors which are symmetric and antisymmetric in overlapping groups

Say I have the following tensor $T_{abc}$ such that $$T_{(a[b)c]}$$ Ergo, it is symmetric in indices $a$ and $b$ and antisymmetric in $b$ and $c$. Keeping in mind the various properties that ...
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### Is there an “internal” definition of the tensor product?

We have the following "internal" definition of the direct sum: A vector space $V$ with subspaces $S,T$ is said to be the direct sum of $S$ and $T$ if $S + T = V$ and $S \cap T = \{0\}$. (Of course ...
2answers
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### Is $a_i\mathbf e^i$ always equal to $a^i\mathbf e_i$?

The way that the covariant basis was described to me was that we could represent any vector $\mathbf a$ as either $\mathbf a=a_i\mathbf e^i$ or $\mathbf a = a^i\mathbf e_i$ (with the Einstein ...
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### Resources on exterior algebra, wedge product, geometric product and tensors

I would like to learn exterior algebra, wedge product and geometric product along with their applications in physics. Is there a good source you can recommend? Should I study differential geometry in ...
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223 views

### Tensor Notation Upper and Lower Indices

I want to ask what the difference between the tensors $T_i^{\; j}$ , $T_j^{\; i}$ , $T_{\; i}^{ j}$ , and $T_{\;i}^{j}$ are. In particular I am asking about the matrix representations of these tensors ...
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### Pullbacks of symmetric tensors commute with products

The problem: Show that $$F*(AB)=(F*A)(F*B)$$ where F is a smooth map from a smooth manifold M to another smooth manifold N, A and B are symmetric tensor fields on N, and $F*$ denotes the pullback ...
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247 views

### Matrix inversion via Levi-Civita symbols

Using Cramer's formula for the inverse of a matrix $M_{ij}$, is it possible to express the entries $(M^{-1})_{ij}$ in terms of the entries $M_{ij}$ using the Levi-Civita symbol and Kronecker deltas? ...
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### Normalize tensors ($3$-by-$3$ matrix) so that the largest eigenvalue is 1?

I am trying to "normalize" a tensor $T$ (a $3$-by-$3$ matrix). The paper says ... the normalization of a tensor scales all eigenvalues so that the largest one equals to $1$. I am confused. ...
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### How are these definitions of the inertia tensor the same?

I'm looking for some help in understanding the inertia tensor (not the physics, just the math). I'm trying to figure out how to convert between the wedge product and tensor product definitions. ...
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### An extension of change of variables in double (and $n$-?) integrals - second-order Jacobian?

I'm aware that there are many, many questions regarding changing variables in double and triple integrals. The equation that typically pops up in textbooks is \begin{align} ...
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