Thread: tensor and einstein summation notation

Hello,

I have some basic problems with tensors and Einstein summation notation. As I understand, if a suffix appears twice in a tensor then summation over repeated indices is assumed, i.e.
,
where is a second order tensor and is the repeated suffix. Is indirectly a zeroth order tensor, i.e. a scalar?

Einstein summation convention also states that a suffix cannot appear more than twice in an expression. I came across an expression for fourth order tensor , where it is mentioned that is a scalar. Why is a zeroth order tensor, i.e. a scalar? I would not expect that since two different suffices are used.

Originally Posted by amrasa81

Hello,

I have some basic problems with tensors and Einstein summation notation. As I understand, if a suffix appears twice in a tensor then summation over repeated indices is assumed, i.e.
,
where is a second order tensor and is the repeated suffix. Is indirectly a zeroth order tensor, i.e. a scalar?

Einstein summation convention also states that a suffix cannot appear more than twice in an expression. I came across an expression for fourth order tensor , where it is mentioned that is a scalar. Why is a zeroth order tensor, i.e. a scalar? I would not expect that since two different suffices are used.

When you sum over a repeated suffix, it reduces the order of the tensor by 2. For the fourth order tensor , when you sum over i the order goes down to 2, then when you sum over j it goes down to 0, so you are left with a scalar.

For a simple example, a square matrix with entries is a second-order tensor. The sum is the trace of the matrix, which is a scalar.