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====== 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 [[wp>Ricci_calculus|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 **co**vector (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).
|}