Overview
The concept of differentiation can be extended to functions that contain more than one variable.
{% f= f(x_1,x_2,...,x_n) %}
There are various concepts of differentiation that can be applied in this case.
The definition of
total derivative
encapsulates the the notion of derivative given for a single variable that also applies to
multi-variable functions.
Alternative to Multi Variable Functions
Multi-variable functions such as
{% f= f(x_1,x_2,...,x_n) %}
can be recast in an equivalent form, as a function of a single variable. In particular, the set of variables can
be collected as a
vector
{% \vec{v} = (x_1,x_2,...,x_n) = \sum_i x^i \vec{e}_i %}
where {% e_1,e_2,...,e_n %}
are the
basis vectors
of the space where {% \vec{v} %} resides.
The function can be recast as
{% f(\vec{v}) %}