- #1
player1_1_1
- 114
- 0
Hell
let make a metric tensor, let it make simple, for random 2-dimentional curved space, ex.
[tex]g_{jk}=\begin{bmatrix}R&0\\ 0&R\sin\phi\end{bmatrix}[/tex]
where R is constant, and [tex]\phi,\varphi[/tex] are variables. Now I have symbol [tex]g_{jk,l}[/tex], does it mean just partial derivative
[tex]\frac{\partial g_{jk}}{\partial x_l}[/tex]?
lets choose [tex]g_{22}[/tex] element and symbol [tex]g_{22,\phi}[/tex]. does it mean
[tex]\frac{\partial g_{22}}{\partial\phi}=R\cos\phi[/tex]?
and the last: in riemann curvature tensor definition is fragment:
[tex]\frac{\partial\Gamma^a_{bc}}{\partial x^d}[/tex]
is it just partial derivative of Christoffel simbol? don't I have to use covariant derivative? thanks for answer!
let make a metric tensor, let it make simple, for random 2-dimentional curved space, ex.[tex]g_{jk}=\begin{bmatrix}R&0\\ 0&R\sin\phi\end{bmatrix}[/tex]
where R is constant, and [tex]\phi,\varphi[/tex] are variables. Now I have symbol [tex]g_{jk,l}[/tex], does it mean just partial derivative
[tex]\frac{\partial g_{jk}}{\partial x_l}[/tex]?
lets choose [tex]g_{22}[/tex] element and symbol [tex]g_{22,\phi}[/tex]. does it mean
[tex]\frac{\partial g_{22}}{\partial\phi}=R\cos\phi[/tex]?
and the last: in riemann curvature tensor definition is fragment:
[tex]\frac{\partial\Gamma^a_{bc}}{\partial x^d}[/tex]
is it just partial derivative of Christoffel simbol? don't I have to use covariant derivative? thanks for answer!