# Notation and conventions used on this site

Every textbook, website, professor, etc. uses their own notation and conventions. To avoid confusion I thought I'd write down mine and try to use them consistently throughout the site.

## Vectors

Notation Meaning  A vector. The norm of the vector (a scalar), if it appears in the text. Otherwise, it's some scalar. The inner product of a vector with itself (a scalar). Equivalently, the norm of the vector squared. The norm of a vector expression (a scalar). The inner product of two vectors (a scalar). The cross product of two vectors (a vector).

## Matrices

Notation Meaning A matrix. A matrix product (a matrix). The former is premultiplying the latter or, equivalently, the latter is postmultiplying the former. A matrix premultiplying a vector or, equivalently, operating on it (a vector). The Kronecker product of two matrices (a matrix). The inverse of an invertible square matrix (a matrix). The determinant of a square matrix (a scalar). The trace of a square matrix (a scalar).

## Index notation

Generally I avoid using indices because of the inherent complexity associated with coordinates. When I do, though, I try to be consistent with Ricci calculus. Below is a quick summary:

Notation Meaning
A superscript. A contravariant index.
A subscript. A covariant index.  The i-th coordinate of a vector (which transforms contravariantly).
I use round brackets to avoid confusion if necessary (in which case the index appears outside). The i-th coordinate of a covector (which transforms covariantly). An element of a (1,1)-tensor.
When the tensor is treated as a linear transformation, this is the element of the corresponding matrix that lies at the intersection of the i-th row and the j-th column.
If and are undefined then this denotes the entire tensor.

## Vector calculus

Notation Meaning A vector field. These usually have uppercase symbols. The gradient of a scalar field (a vector field). The divergence of a vector field (a scalar field). The curl of a vector field (a vector field). The Laplacian of a scalar field (a scalar field). 