Semicolon notation in component of covariant derivative

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• berlinspeed
In summary, the semicolon is used to denote the covariant derivative of a tensor, which can be expressed in index or abstract index notation as the sum of its components multiplied by the respective basis vectors and co-vectors. This notation is useful for compactly representing tensors and their derivatives. To type equations, you can use PF's LaTeX feature.
berlinspeed
TL;DR Summary
The use of semicolon notation as covariant derivative
Can someone clarify the use of semicolon in

I know that semicolon can mean covariant derivative, here is it being used in the same way (is
expandable?) Or is
a compact notation solely for the components of
?

If you're familiar with index or abstract index notation

$$S^\alpha{}_{\beta\gamma;\delta} = \nabla_\delta \, S^{\alpha}{}_{\beta\gamma}$$

berlinspeed said:
Can someone clarify the use of semicolon in
View attachment 245677
I know that semicolon can mean covariant derivative, here is it being used in the same way (is View attachment 245678 expandable?) Or isView attachment 245679 a compact notation solely for the components ofView attachment 245680?
Not sure what your confusion is. Any tensor can be written as the sum of each of its components in some basis multiplied by the outer product of the respective basis vectors and co-vectors. Since the covariant derivative of a tensor is still a tensor (but one rank higher), it can also be written this way. Does that clear anything up for you?

pervect said:
If you're familiar with index or abstract index notation

$$S^\alpha{}_{\beta\gamma;\delta} = \nabla_\delta \, S^{\alpha}{}_{\beta\gamma}$$
Hello thank you, can you tell me what did you use to type your equations? I'm new to this site and trying to find a good way to generate equations. Much appreciated.

vanhees71, JD_PM and berkeman
vanhees71 and JD_PM

1. What is semicolon notation in the component of covariant derivative?

Semicolon notation in the component of covariant derivative is a mathematical notation used in differential geometry to represent the components of a covariant derivative operator. It is typically written as a semicolon between two indices, such as Di;j, and is used to denote the partial derivative of a vector field with respect to one index and the covariant derivative with respect to the other index.

2. How is semicolon notation different from comma notation in the component of covariant derivative?

Semicolon notation and comma notation in the component of covariant derivative are two different ways of representing the same mathematical concept. While semicolon notation is used to denote the partial derivative and covariant derivative, comma notation is used to denote the partial derivative and the Christoffel symbols. Both notations are commonly used in differential geometry and have their own advantages and applications.

3. What is the significance of semicolon notation in the component of covariant derivative?

Semicolon notation is significant because it allows for a compact and concise representation of the covariant derivative operator, making it easier to work with in mathematical calculations. It also helps to distinguish between the partial derivative and the covariant derivative, which have different geometric interpretations.

4. Can semicolon notation be used in other areas of mathematics?

Yes, semicolon notation can be used in other areas of mathematics, such as in tensor calculus and differential equations. It is a commonly used notation in differential geometry, but its applications can extend to other fields as well.

5. Are there any limitations or drawbacks to using semicolon notation in the component of covariant derivative?

One limitation of semicolon notation is that it can become cumbersome when dealing with higher order derivatives or tensors with more than two indices. In these cases, alternative notations may be used. Additionally, semicolon notation may be confusing for those who are not familiar with it, so it is important to clearly define and explain its use in mathematical discussions.

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