All Questions
Tagged with vector-analysis tensors
144
questions
25
votes
7
answers
11k
views
How would you explain a tensor to a computer scientist?
How would you explain a tensor to a computer scientist? My friend, who studies computer science, recently asked me what a tensor was. I study physics, and I tried my best to explain what a tensor is, ...
9
votes
2
answers
2k
views
Special case of the Hodge decomposition theorem
I am trying to prove the following special case of the Hodge decomposition theorem in differential geometry for an $n$ component vector field $V_i$ in $\mathbb{R}^n$. I have very little knowledge of ...
user23238
7
votes
2
answers
289
views
What is $\left ( \vec{\nabla} \times \vec{A} \right ) \cdot \left ( \vec{\nabla} \times \vec{A} \right )$?
I'm trying to rewrite $\left ( \vec{\nabla} \times \vec{A} \right ) \cdot \left ( \vec{\nabla} \times \vec{A} \right )$ in some other way. I tried using Levi-Civita symbol and Kronecker delta, but I'm ...
- 120
7
votes
2
answers
2k
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vector/tensor covariance and contravariance notation
As I looked over the Wikipedia article on covariance and contravariance of vectors and $\mathbf{v}=v^i\mathbf{e}_i$ is said as a contravariant vector while $\mathbf{v}=v_i\mathbf{e}^i$ is said as ...
- 905
7
votes
1
answer
3k
views
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 ...
- 123
6
votes
1
answer
8k
views
How can you calculate distance in plane polar coordinates using the metric tensor
I'm trying to develop some intuition about the metric tensor and how it can be used to calculate distances/angles in curvlinear space by using the very simple example of a 2D polar surface.
The ...
- 63
6
votes
2
answers
8k
views
Formula of the gradient of vector dot product
On Wikipedia in the article "Vector calculus identities"
(https://en.m.wikipedia.org/wiki/Vector_calculus_identities)
there are the following two formulas for computing the gradient of vector dot ...
- 61
6
votes
1
answer
467
views
Net torque on a surface in Stokes flow
I'm having difficulty seeing how the net torque on a surface in Stokes flow is non-zero.
For Stokes flow, we have $\nabla\cdot\sigma = 0$, where $\sigma$ is a symmetric stress tensor. The net torque ...
6
votes
1
answer
269
views
Geometric meaning of interior,exterior derivatives, and, Hodge Duality
I've been tinkering with differential forms for a while now, and I've had a few questions all rolled into one trying to understand them. The exterior derivative is quite natural to me - it looks just ...
- 17.9k
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
5
votes
3
answers
259
views
What are the linear transformations that preserves the cross product, i.e. $ R(v\times w) = (Rv) \times (Rw), \forall v,w \in \mathbb{R}^3 $
Let us just focus on $\mathbb{R}^3$ currently. We study the set of all $3\times 3$ matrices $R$ satisfying
$$
R(v\times w) = (Rv) \times (Rw), \forall v,w \in \mathbb{R}^3
$$
where $\times$ is the ...
- 658
5
votes
2
answers
495
views
Notation: $\nabla \cdot$, div, $\nabla$, grad, ...? [closed]
I am currently finding myself doing lots of applied mathematics, e.g. fluid dynamics, and of course this involves a lot of vector calculus amongst other things. This had me thinking about proper ...
5
votes
1
answer
319
views
What is the most general/abstract way to think about Tensors
In their most general and abstract definitions as Mathematical Objects :
A Scalar is an element of a field used to define Vector Spaces
A Vector is an element of a Vector Space.
Since a Scalar is ...
- 13.2k
5
votes
1
answer
74
views
Is this 2-tensor symmetric? It satisfies these conditions
I have some scalar field $p$ and a 2nd order tensor $\textbf{T}$ such that
$div( \ \textbf{T} \ \textbf{grad}(p) \ )=0$
$\textbf{curl}(\ \textbf{T} \ \textbf{grad}(p) \ )=\textbf{0} $
$\textbf{T}$ ...
- 167
4
votes
3
answers
6k
views
Divergence of the product of a tensor and a vector field
Let $\mathbf u$ and $\mathbf S$ be smooth fields with $\mathbf u$ vector valued and $\mathbf S$ tensor valued. I would like to prove the following identity:
$$\operatorname{div}\mathbf S\mathbf{u}=\...
- 1,535