Notation
[This page is under construction].
- We use Einstein’s summation convention.
- We adopt the metric signature
.
- We generally adopt the notation of [MTW73], unless explicitly stated otherwise.
- Tensors will be treated either by “index free” (geometrical) notation or “classical component” (index) notation, or both, depending on the point that is being addressed/emphasized. Tags denoted “Index-Free” and “Full-Index” will be used to indicate appropriately.
- In our initial discussions on tensors, you will also see that we will be using an unorthodox notation for tensors, particularly for Euclidean tensors. Given an Euclidean tensor
of rank
, it has
input slots (operates on
) and outputs another tensor of rank
. We write the tensor
. That notation will be dropped later.
- We adopt any of the following notations for the Cartesian (orthonormal) basis vectors and for the components of a vector
in that basis, respectively:
,
,
,
,
,
.
- A tensor
of type
at a point
of a manifold is a linear function that takes as its arguments
one-forms and
vectors. We also write it as
.
