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
$ \mathbf{x} $
$ \mathbf{x_\textup{stuff}} $
A vector.
$ x $ The norm of the vector $\textbf{x}$ (a scalar), if it appears in the text. Otherwise, it's some scalar.
$ x^2 $ The inner product of a vector with itself (a scalar). Equivalently, the norm of the vector squared.
$ \vert\mathbf{a} + \mathbf{b} \cross \mathbf{c}\vert $ The norm of a vector expression (a scalar).
$ \mathbf{a} \cdot \mathbf{b} $ The inner product of two vectors (a scalar).
$ \mathbf{a} \cross \mathbf{b} $ The cross product of two vectors (a vector).

Matrices

Notation Meaning
$ \hat{M} $ A matrix.
$ \hat{A}\hat{B} $ A matrix product (a matrix). The former is premultiplying the latter or, equivalently, the latter is postmultiplying the former.
$ \hat{M}\mathbf{x} $ A matrix premultiplying a vector or, equivalently, operating on it (a vector).
$ \hat{A} \otimes \hat{B} $ The Kronecker product of two matrices (a matrix).
$ \hat{M}^{-1} $ The inverse of an invertible square matrix (a matrix).
$ \textup{det}~\hat{M} $ The determinant of a square matrix (a scalar).
$ \textup{Tr}~\hat{M} $ 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.
$ \mathbf{x}^i $
$ (\mathbf{x}_\textup{stuff})^i $
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).
$ \mathbf{x}_i $ The i-th coordinate of a covector (which transforms covariantly).
$ \tensor{T}{_i^j} $ 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 $i$ and $j$ are undefined then this denotes the entire tensor.

Vector calculus

Notation Meaning
$ \mathbf{A} $ A vector field. These usually have uppercase symbols.
$ \nabla \psi $ The gradient of a scalar field (a vector field).
$ \nabla \cdot \mathbf{A} $ The divergence of a vector field (a scalar field).
$ \nabla \cross \mathbf{A} $ The curl of a vector field (a vector field).
$ \Delta \psi $ The Laplacian of a scalar field (a scalar field).