Where possible, vector quantities are displayed with a raised arrow (for example,
, 
). Boldfaced characters
are reserved for vectors and matrices as they apply to linear algebra
(for example, the identity matrix, 
). The operator
, referred to as grad, nabla,
or del, represents the partial derivative of a quantity with respect
to all directions in the chosen coordinate system. In Cartesian coordinates, 
is defined to be 
(1–1)
appears in several ways: The gradient of a scalar quantity is the vector whose components are the partial derivatives; for example,
(1–2)
The gradient of a vector quantity is a second-order tensor; for example, in Cartesian coordinates,
(1–3)
This tensor is usually written as
(1–4)
The divergence of a vector quantity, which is the inner product between
and a vector; for example,
(1–5)
The operator
, which
is usually written as 
and
is known as the Laplacian; for example, 
(1–6)
is different
from the expression 
, which is defined as 
(1–7)
An exception to the use of
is
found in the discussion of Reynolds stresses in Advanced Turbulence Models, where convention dictates the use
of Cartesian tensor notation. In this section, you will also find
that some velocity vector components are written as u, v, and w instead of the
conventional v with directional subscripts.