Tensor Notation Explained - Understanding Differences and Derivatives

[Old Guy]

Homework Statement

Not a specific problem; I'm trying to understand what the notation means; I'm using primarily Griffiths, Marion and Jackson textbooks.

The notation for a matrix, with the superscript row index and subscript column index I understand. For the EM field tensor, Griffiths has both indices superscripted, and Marion has both subscripted. My questions are:
1. What is the difference?
2. What is the meaning when they are both on the same level? I assume one is rows and one is columns - but which is which?
3. When you take a derivative of the tensor, what exactly are you doing in terms of the indices of the tensor and the index of the variable of integration?

Homework Equations

The Attempt at a Solution

I've searched Boas, Arfken & Webber, Wikipedia, and some other web sites to no avail.

 

[gabbagabbahey]

Homework Statement

Not a specific problem; I'm trying to understand what the notation means; I'm using primarily Griffiths, Marion and Jackson textbooks.

The notation for a matrix, with the superscript row index and subscript column index I understand. For the EM field tensor, Griffiths has both indices superscripted, and Marion has both subscripted. My questions are:
1. What is the difference?

The difference is in how the components of the tensor transform, under any given coordinate transformation. The usual convention is that tensors with lowered (subscript) indices are covariant and tensors with raised (superscript) indices are contravariant. Tensors with both lowered and raised indices are called mixed tensors.

2. What is the meaning when they are both on the same level? I assume one is rows and one is columns - but which is which?

Tensors are not just matrices. In the case of second rank tensors (whether they are covariant, contravariant or mixed), you can represent them by a matrix by defining certain basis vectors to be represented as row vectors, and others as columns. Which is which depends on how you define your representation.

3. When you take a derivative of the tensor, what exactly are you doing in terms of the indices of the tensor and the index of the variable of integration?

Taking the derivative of a tensor with respect to any given variable simply means that you take the derivative of each component of the tensor. In the case of the field tensors you work with in electrodynamics, there are 16 components for each second rank tensor (matrix) and 4 components for each 1st rank tensor (vector).