Differences in Presentation of Ordinary Partial Derivatives of Tensors

[mokrunka]

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?

 

[grzz]

see attachment

 

[mokrunka]

Right, I know they're the same from the chain rule.

My question really I guess is why is it necessary to add the additional term? Is it because$A_{r}(x^{p}(x^{k'}))$?

Since the text seems to present this in different ways, I'd really like to understand the mechanics of why this extra term is there (and then cancelled) so that I can make sure I understand the derivations and tensor operations further along.

 

[grzz]

You are guessing correctly.

 

[sardar]

I can provide some insight into the differences in the presentation of ordinary partial derivatives of tensors. It is important to note that there are different conventions and notations used in the field of relativity and tensors, and this can lead to variations in how the ordinary partial derivative is presented.

In the first example, the ordinary partial derivative is presented as a simple fraction with the numerator being the derivative of one tensor component with respect to another, and the denominator being the original tensor component. This notation is commonly used in many textbooks and is a straightforward way to represent the derivative.

In the second example, an additional partial derivative is included in the second term of the right-hand side of the equation. This is known as the "chain rule" and is commonly used in mathematics to calculate the derivative of a composite function. In this case, the composite function is the tensor component with respect to the different coordinate systems. This notation is used in some textbooks to emphasize the importance of the chain rule in understanding the covariant derivative.

The differences in presentation may also be due to the level of mathematical rigor and complexity in the textbook. Some textbooks may choose to omit the additional partial derivative for simplicity, while others may include it to fully explain the concept of the covariant derivative.

Ultimately, it is important to understand the underlying principles and concepts behind the notation used in different textbooks. As you continue to study relativity and tensors, you will likely encounter different notations and presentations, but as long as you have a solid understanding of the concepts, you will be able to navigate through them successfully.