All Questions
3
questions
106
votes
4
answers
10k
views
Why does nature favour the Laplacian?
The three-dimensional Laplacian can be defined as $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}.$$ Expressed in spherical coordinates, it ...
- 1,327
2
votes
2
answers
282
views
Is the contracted Christoffel symbol a tensor?
The coordinate transformation law (from coordinates x to coordinates y) for the Christoffel symbol is:
$$\Gamma^i_{kl}(y)=\frac{\partial y^i}{\partial x^m} \frac{\partial x^n}{\partial y^k} \frac{\...
- 633
0
votes
3
answers
991
views
Christoffel symbol and covariant derivative
I came across the Christoffel symbols via the geodesic equation, and I understand the extrinsic form and the intrinsic form and can prove that they are identical:
extrinsic form:
$$\Gamma^{j}_{~ik}=\...
- 157