All Questions
3
questions
0
votes
1
answer
59
views
Metric Tensor Grid
Let, we are in a 2d metric where $g_{xx}=1, g_{yy}=x^2$, therefore $|e_x|=1$, $|e_y|=x$. If we try to draw the metric in a grid - it looks something like the image I uploaded. Note that, along the X ...
0
votes
0
answers
61
views
Confusion between covariant and partial derivatives
Let, we are in 1d cartesian space with metric $g_{xx} = x^2$. Let we have a vector $v = 1/x e_x$. Since the vector is designed to shrink its components as the basis grows - its total length will ...
0
votes
1
answer
143
views
Tensor calculus - product of metric tensor and second covariant derivative of a scalar (Laplace-Beltrami operator)
I am trying to prove the following.
Suppose we have a scalar function $\phi$ (sufficiently differentiable), the metric tensor $g_{ij} = \dfrac{\partial y^\alpha}{\partial x^i}\dfrac{\partial y^\alpha}{...