Table of Contents
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 ith coordinate of a vector (which transforms contravariantly). I use round brackets to avoid confusion if necessary (in which case the index appears outside). 
The ith 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 ith row and the jth 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). 