All Questions
Tagged with vector-analysis tensors
35
questions with no upvoted or accepted answers
6
votes
0
answers
207
views
'Tensor Calculus' by J.L. Synge and A. Schild (1979 Dover publication) . Exercise 12 page 110.
I'm solving all the exercises of 'Tensor Calculus' by J.L. Synge and A. Schild (1979 Dover publication) . Till now everything went smooth, but now I'm stuck at exercise 12. page 110 of the third ...
- 116
4
votes
0
answers
101
views
What is a neat way to solve $\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{\mathbf{C}}$?
Let $\mathbf{u}:\mathbb{R}^3\to\mathbb{R}^3$ be a smooth enough vector field that satisfies the following equation
$$\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{C},\tag{1}$$
where $\nabla\mathbf{u}$ ...
- 15k
4
votes
1
answer
291
views
Divergence operator of higher order and intrinsic point of view
Let $\underline{u}$ be a $1$ - order tensor (say a column vector) I want to prove that :
$\underline{\operatorname{div}} \left( (\underline{\underline{\operatorname{grad}}} \, \underline{u})^T\...
- 1,294
3
votes
0
answers
250
views
Relation between the curl of a vector field and the divergence of a tensor
The following seemingly-simple problem came up when working on a problem in the fluid theory of plasmas.
Given a vector field $\mathbf{A}$, find a symmetric tensor $\mathbf{P}$ such that $\...
- 22.9k
3
votes
0
answers
1k
views
Gradient is covariant or contravariant?
I read somewhere people write gradient in covariant form because of their proposes.
I think gradient expanded in covariant basis i , j , k so by invariance nature of vectors, component of gradient ...
user87128
2
votes
0
answers
445
views
Show that the derivative of a second order tensor gives a third order tensor
Let $U_{i,j}$ be a second order tensor. Show that $\frac{\partial U_{i,j}}{\partial x_{k}}$ is a third order tensor.
I know how to prove that the gradient of a scalar field (which is a tensor of ...
- 21
2
votes
1
answer
505
views
Vectors, Forms, Multivectors, and Tensors
In researching some of the ways that vectors (and vector fields) generalize I find that there are apparently many different objects that generalize them -- matrices, differential forms/ covectors, ...
- 259
1
vote
1
answer
140
views
Covariant and partial derivative of a vector field (not component)
Is the covariant derivative of a vector field (not the components of a vector) same as the partial derivative?
I am adding a screenshot from page 69 from General Relativity: An introduction for ...
- 109
1
vote
0
answers
40
views
Whats the significance of $g^{-1}$ (the inverse metric) appearing in tangential projection?
Let $M \subseteq (\mathbb{R}^n,g_E)$ be an embedded submanifold, with the embedding $F : M \to \mathbb{R}^n$. It is well known (c.f. Lee, doCarmo) that the covariant derivative on $M$ with respect to ...
- 198
1
vote
0
answers
80
views
Calculate the angle between vectors in equation. When does the conductivity tensor component take the form $\sigma_{ab} = \bar \sigmaδ_{ab}$?
In a certain anisotropic conductive material, the relationship between the current density $\vec j$ and
the electric field $\vec E$ is given by: $$\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\...
- 23
1
vote
0
answers
72
views
What is the definition of $\nabla \cdot(\vec{x}\times\textbf{T})$?
Consider a vector $\vec{x}$ in three dimensions and $3\times 3$ second rank symmetric tensor $\textbf{T}$.
What is the definition of $\nabla \cdot(\vec{x}\times\textbf{T})$?
Based on this answer, I ...
- 2,424
1
vote
0
answers
48
views
A problem with index notation. Related variational calculus.
i was reading a paper and i find something that i don't understand.
This is the paper "An action principle for action-dependent Lagrangians: Toward an action principle to non-conservative systems".
...
1
vote
0
answers
466
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Understanding role of tangent/cotangent space changes under coordinate transformations
This is kind of a follow-up to the excellent answer to this question: https://physics.stackexchange.com/questions/445948/general-coordinate-transformations
I want a very clear understanding of what ...
- 2,569
1
vote
1
answer
249
views
Proving zero identities in vector calculus with simple arguments involving index counting or symmetry?
Consider the following table describing four second derivative operators.
...
user13618
1
vote
1
answer
643
views
Christoffel Symbols for elliptic coordinate system
Does anyone know the Christoffel symbols of second kind for the elliptic coordinate system:
\begin{matrix}
x = R\cosh(u)\cos(v)\\
y = R\sinh(u)\sin(v)\\
z = z
\end{matrix}
the metric tensor is ...
