- #1

Jianbing_Shao

- 102

- 2

- TL;DR Summary
- differentail equations, parallel transport equations

In genaral relativity, how to solve differential equations is seldom be discussed. I want to know how to sole the differential equations like this:

$$\partial_kv^i(x)+\Gamma^i_{jk}(x)v^j(x)=\partial_kA^i(x)$$

Here ##\Gamma^i_{jk}(x)## is connection field on a manifod and ##A^i(x)## is a vector field on the manifold. then how to get ##v^i(x)##, Perhapa we can not find a global vector field which can satisfy the differential equations, But we can still find a solution on a partilular curve. then how to find it?

$$\partial_kv^i(x)+\Gamma^i_{jk}(x)v^j(x)=\partial_kA^i(x)$$

Here ##\Gamma^i_{jk}(x)## is connection field on a manifod and ##A^i(x)## is a vector field on the manifold. then how to get ##v^i(x)##, Perhapa we can not find a global vector field which can satisfy the differential equations, But we can still find a solution on a partilular curve. then how to find it?