All Questions
Tagged with vector-analysis tensors
144
questions
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Partial derivatives in tensor/index notation!
For my continuum mechanics class, I'm tasked with finding $\nabla u$, $u$ being $u = b\frac{x}{|x|^3}$. Here, $b$ is a scalar constant.
Attempt at the solution:
I rewrite $\frac{1}{|x|^3}$ as $(x_i^2)^...
- 11
2
votes
1
answer
427
views
Proof of expression for Christoffel symbols of the first kind, $[i , j k] = {\bf e}_i \cdot \frac{\partial {\bf e}_j}{\partial x^k}$
On page 155 of Vector and Tensor Analysis with Applications, by A.I Borishenko and I.E. Tarapov, the authors assert that,
$$\frac{\partial {\bf e}_j}{\partial x^k} = \left\{ i \atop j \; k \right\} {\...
- 1,261
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0
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80
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Calculate the angle between vectors in equation. When does the conductivity tensor component take the form $\sigma_{ab} = \bar \sigmaδ_{ab}$?
In a certain anisotropic conductive material, the relationship between the current density $\vec j$ and
the electric field $\vec E$ is given by: $$\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\...
- 23
1
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1
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How should I calculate fourth order tensor times second order tensor?
Let's say I have two second-order tensors
${\mathbf{S}} = {S_{ij}}{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_j}$
and
${\mathbf{T}} = {T_{ij}}{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_j}$
. Then, I know
${\...
- 11
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1
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149
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Tensor Calculus Notation
In tensor notation, we know the following is true for general vectors:
$$
\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C}) = A_i\epsilon_{ijk}B_jC_k = -B_j\epsilon_{jik}A_iC_k
$$
However, if we try and ...
- 133
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1
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58
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Show that $ \nabla . ({\bf T}^T {\bf v}) = {\bf T} : \nabla {\bf v} + {\bf v} \cdot (\nabla . {\bf T})$
Here is my feeble attempt:
$$
\begin{equation}\begin{aligned}
\nabla .({\bf T}^T {\bf v}) & = \frac{\partial {\bf T}_{ji}}{\partial x_j} {\bf v}_i + {\bf T}_{ji} \frac{\partial {\bf v}_{i}}{\...
- 1,261
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0
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223
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Gradient of a vector field in curvilinear coordinates
What is the formula for calculating the gradient of a vector field if the field is expressed in terms of curvilinear coordinates like spherical or cylindrical systems?
0
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34
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Matrix differential equation set to zero
I have a bijective continuous function $f$, which maps an $(n\times1)$ dimensional column vector $t=[t_1,...t_n]'$ to another $(n\times1)$ dimensional column vector $f(t)=[f_1(t),...f_n(t)]'$. I ...
- 766
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111
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Finding $\mathbf{u} \cdot \nabla \mathbf{u}$ in cylindrical coordinates
Evaluate $\mathbf{u}\cdot\nabla\mathbf{u}$ (the directional derivative of $\mathbf{u}$ in the direction of $\mathbf{u}$)in cylindrical coordinates $(r, \phi,z)$, where $\bf{u}=e_{\phi}$.
The textbook ...
- 927
0
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1
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119
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Index notation: associative property
I have given the following term, that I can write in index notation as following:
$$ (\mathbf{a} \cdot \nabla)\mathbf{b} = a_j \partial_j b_i$$
Now I can exchange the order and get
$$ \partial_j b_i ...
- 163
1
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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
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48
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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
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0
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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
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914
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vector analysis: prove Cartesian gradient of invariant scale field is also invariant under rotation of axis
I don't know how to explain where i'm stuck without explaining most of the exercise... so here it is:
BACKGROUND
Given two rectangular coordinate systems that having the same origin but the axes are ...
- 971
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0
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23
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Identity relating $V \circ \nabla F$ and $V( \nabla \circ F)$?
I’ve been trying to derive the Leibniz Integral Rule 1 in 3D and I’ve nearly got it, but where Wikipedia shows $V \circ \nabla F$, my approach gives $V( \nabla \circ F)$. I’m thinking that the two ...
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