- #1
mokrunka
- 5
- 0
Ok folks, I've taken a stab at the Latex thing (for the first time, so please bear with me).
I've mentioned before that I'm teaching myself relativity and tensors, and I've come across a question.
I have a few different books that I'm referencing, and I've seen them present the ordinary partial derivative in 3 different ways. It took me an hour to put together what I have below, so I have omitted the 3rd presentation, but I think I'm getting the message across with the examples below.
1)
V_{k'}=\frac{\partial u^{i}}{\partial u^{k'}}V_{k}
\frac{\partial}{\partial u^{l'}}(\frac{\partial u^{i}}{\partial u^{k'}}V_{i})=\frac{\partial^{2}u^{i}}{\partial u^{l'}\partial u^{k'}}V_{i}+\frac{\partial V_{i}\partial u^{i}}{\partial u^{l'}\partial u^{k'}}
2)
A_{s'}=\frac{\partial x^{r}}{\partial x^{s'}}A_{r}
\frac{\partial A_{s'}}{\partial x^{k'}}=\frac{\partial^{2}x^{r}}{\partial x^{k'}\partial x^{s'}}A_{r}+\frac{\partial A_{r}\partial x^{p}\partial x^{r}}{\partial x^{p}\partial x^{k'}\partial x^{s'}}
In the second example, there seems to be an additional partial derivative being conducted in the second term on the RHS of the equation. This is the first step (in the text) of explaining the need for the covariant derivative, and I'm trying to understand why some texts have this additional partial derivative, and indeed the third text has 3 sets of partial derivatives in the first term of the RHS of the equations. Can anyone help me understand why this term is here, and why the presentation is inconsistent?
I've mentioned before that I'm teaching myself relativity and tensors, and I've come across a question.
I have a few different books that I'm referencing, and I've seen them present the ordinary partial derivative in 3 different ways. It took me an hour to put together what I have below, so I have omitted the 3rd presentation, but I think I'm getting the message across with the examples below.
1)
V_{k'}=\frac{\partial u^{i}}{\partial u^{k'}}V_{k}
\frac{\partial}{\partial u^{l'}}(\frac{\partial u^{i}}{\partial u^{k'}}V_{i})=\frac{\partial^{2}u^{i}}{\partial u^{l'}\partial u^{k'}}V_{i}+\frac{\partial V_{i}\partial u^{i}}{\partial u^{l'}\partial u^{k'}}
2)
A_{s'}=\frac{\partial x^{r}}{\partial x^{s'}}A_{r}
\frac{\partial A_{s'}}{\partial x^{k'}}=\frac{\partial^{2}x^{r}}{\partial x^{k'}\partial x^{s'}}A_{r}+\frac{\partial A_{r}\partial x^{p}\partial x^{r}}{\partial x^{p}\partial x^{k'}\partial x^{s'}}
In the second example, there seems to be an additional partial derivative being conducted in the second term on the RHS of the equation. This is the first step (in the text) of explaining the need for the covariant derivative, and I'm trying to understand why some texts have this additional partial derivative, and indeed the third text has 3 sets of partial derivatives in the first term of the RHS of the equations. Can anyone help me understand why this term is here, and why the presentation is inconsistent?