Partial derivative
A partial derivative of a function of many variables is the derivative of that function with respect to one of its arguments.
For example, if then
has three partial derivatives at the point
:
Del operator
The del operator, or nabla symbol, written , represents the vector
where the value
is the arity (number of arguments) of the function in question.
Gradient
The product of and a function
is
a vector storing, in order, all of the partial derivatives of
.
The gradient applies when is a scalar-valued function of many variables. For example, the gradient of temperature in a closed room is
, where
,
, and
are the Cartesian coordinates in the three spatial dimensions: length, width, and height, respectively.
and
are likely to be close to zero at most points, but
probably has a small positive value, since the air nearer the ceiling (greater
) is warmer than the air nearer the floor (lesser
). Therefore, the typical direction of the gradient vector
is close to upwards.