All Questions
10
questions
2
votes
1
answer
67
views
Differential surface element and nabla operator
If we have the vector field $\vec{u}=\vec{A}\times \vec{v}$, where $\vec{A}=\text{const.}$ and we integrate over some closed curve, by using Stokes' theorem we get:
$$
\begin{align}
\oint_{\partial S}...
- 379
0
votes
0
answers
120
views
How do I simplify $\delta_{ij} \delta^{jk}$?
How do I simplify $\delta_{ij} \delta^{jk}$?
I know that $\delta_{ij} \delta_{jk}=\delta_{ik}$, but what do I do if the there's a Kronecker Delta symbol with upper indices and one with lower indices?
- 93
0
votes
0
answers
34
views
Can you define a tensor by integrating one vector with respect to another?
I was reading this question, simply I was wondering about integrating a vector with respect to another vector field. In the question, the OP asks if the following quantity has any sensible meaning:
$$\...
- 211
0
votes
2
answers
236
views
Power of a second-order tensor
I have the following equation:
$$
\dfrac{\mathbf{V}^T \cdot \mathbf{V}}{\mathbf{V}^T : \mathbf{V}} + \dfrac{\mathbf{V} \cdot \mathbf{V}^T}{\mathbf{V} : \mathbf{V}^T} = \dfrac{\mathbf{D}^2}{\left \| \...
user1012020
0
votes
1
answer
379
views
$C_{ij}=T_{ijklmn} D_{kl} D_{mn}$ where $T_{ijklmn}$ is a rank 6 isotropic tensor, $C_{ij}$ is symmetric and $D_{ij}$ is antisymmetric
I was doing a question on Tensors and hit a roadblock
The Question:
Suppose that $C_{ij}$ and $D_{ij}$ satisfy the quadratic relationship $C_{ij} = T_{ijklmn} D_{kl}D_{mn}$,
where $T_{ijklmn}$ is an ...
- 23
1
vote
1
answer
1k
views
How should I calculate fourth order tensor times second order tensor?
Let's say I have two second-order tensors
${\mathbf{S}} = {S_{ij}}{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_j}$
and
${\mathbf{T}} = {T_{ij}}{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_j}$
. Then, I know
${\...
- 11
0
votes
1
answer
119
views
Index notation: associative property
I have given the following term, that I can write in index notation as following:
$$ (\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
0
votes
0
answers
41
views
Two questions about vector equations
Q1. How to write the following equation in a form of $\vec{a}= something$?
From the question posted in Physics SE, I found the following process is wrong.
For any $ \vec{v}\ne\vec{0}$
$$ \vec{v}\...
- 45
0
votes
0
answers
50
views
How would you express $(\underline{a} \times \underline{b}) \times (\underline{a} \times \underline{c})$ in index notation?
At a guess, I would have said that the answer is $\varepsilon_{ijk}\varepsilon_{jlm}a_{l}b_{m}\varepsilon_{kpq}a_{p}c_{q}$, but I'm guessing that this is incorrect. What is the corrdct expression?
...
- 2,737
4
votes
1
answer
10k
views
Divergence of stress tensor in momentum transfer equation
Let suppose that we work in a 2D cartesian coordinates. what will be x and y components of
$\nabla.\left[-p I+\mu \left(\nabla \text{u}+(\nabla \text{u})^T\right)-\frac{2}{3} \mu (\nabla.\text{u})
I \...
- 4,358