All Questions
5
questions
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
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
4
votes
1
answer
942
views
Divergence theorem in curvilinear coordinates
Suppose I have a tensor
\begin{gather}
\stackrel{\leftrightarrow}{A} =
\begin{bmatrix}
a_{11}(\vec{r}) & a_{12}(\vec{r}) & a_{13}(\vec{r})\\
a_{21}(\vec{r}) & a_{22}(\vec{r}) & a_{23}(...
- 235
3
votes
3
answers
2k
views
What are the dual basis vectors?
What exactly are dual basis vectors such as those which arise in non-orthogonal co-ordinate systems?
What is their physical interpretation.
Please note, I don't know much tensor calculus yet. I am ...
- 1,486
1
vote
1
answer
1k
views
Index/Einstein notation to derive Gibbs/Tensor relations
In a few continuum classes I have seen indicial notation used to derive relations in Gibbs notation. However, Gibbs notation is valid for all coordinates while indicial notation is valid only for ...
- 146