All Questions
7
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:
$$
\begin{align}
\oint_{\partial S}...
- 379
0
votes
0
answers
52
views
Intuition about the divergence of a vector field in non-orthogonal basis
My textbook defines the divergence of a vector field in a non orthogonal constant basis the following way:
$$div(\vec{u})=\vec{a}^i\cdot\frac{\partial \vec{a}_ku^k}{\partial x^i}=\frac{\partial u^i}{\...
- 379
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
4
votes
2
answers
413
views
Proving $A\times(\nabla\times B)+B\times(\nabla\times A)=\nabla(A\cdot B)-(A\cdot\nabla)B -(B\cdot\nabla)A$ with Einstein summation
So, I'm seeking to prove the below identity, for $A,B$ vectors fields in $\mathbb{R}^3$:
$$A \times (\nabla \times B) + B \times (\nabla \times A) = \nabla (A \cdot B) - (A \cdot \nabla)B - (B \cdot \...
- 45.9k
4
votes
0
answers
101
views
What is a neat way to solve $\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{\mathbf{C}}$?
Let $\mathbf{u}:\mathbb{R}^3\to\mathbb{R}^3$ be a smooth enough vector field that satisfies the following equation
$$\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{C},\tag{1}$$
where $\nabla\mathbf{u}$ ...
- 15k
2
votes
1
answer
1k
views
How to compute the divergence in polar coordinates from the Voss-Weyl formula?
The Voss-Weyl formula reads
$$\nabla_\mu V^\mu=\frac{1}{\sqrt g}\partial_\mu(\sqrt g V^\mu),$$
where $g=\mathrm{det}( g_{\mu\nu})$. In polar plane coordinates the only non-vanishing components of the ...
- 21
0
votes
1
answer
762
views
A question about vector fields and divergence
I am reading the paper http://www.goshen.edu/physix/mathphys/gco/TensorGuideAJP.pdf in order to gain a basic understanding about tensors. I had some difficulties about understanding some definitions.
...
- 3,036