Partial derivatives


[2. Mathematical elements.]Partial derivatives.
Let φ (x, y) be a complex function of two real variables.
As an example, let
φ (x, y) = e^{ i x y}.
There are two different rates of change of this function, one with respect to the
x variable, the other with respect to the y variable.
The first rate of change is determined considering the y variable as a
constant and calculating the usual derivative of φ .
This is called the first partial derivative of φ (x, y) with
respect to x and is denoted by
∂φ (x, y) / ∂x.
In the example we have
∂φ (x, y) / ∂x = ∂e^{ i x y} / ∂x = y e^{ i (xy + π/2)} .
A simple example: let z = x^{3} y. We have:
Note that, in the example, ∂²z / ∂x ∂y = ∂²z / ∂y ∂x. In this article are mainly used function satisfying this relation, although it does not hold in general.
