Multi Variable Differentiation
Overview
The concept of differentiation
{% f= f(x_1,x_2,...,x_n) %}
Jacobian
The
matrix representation
of the total derviative is called the Jacobian and is calculated as follows:
{%
\frac{\partial{\vec{f}}}{\partial{\vec{x}}} =\begin{bmatrix}
\frac{\partial{f_1}}{\partial{x_1}} & \frac{\partial{f_1}}{\partial{x_2}} & ... & \frac{\partial{f_1}}{\partial{x_n}} \\
\frac{\partial{f_2}}{\partial{x_1}} & \frac{\partial{f_2}}{\partial{x_2}} & ... & \frac{\partial{f_2}}{\partial{x_n}} \\
... \\
\frac{\partial{f_m}}{\partial{x_1}} & \frac{\partial{f_m}}{\partial{x_2}} & ... & \frac{\partial{f_m}}{\partial{x_n}} \\
\end{bmatrix}
%}
Change of Variables
The Jacobian is used in the change of variables formula for integration
(shown here for 2 variables)
{% \int \int f(x,y) dx dy = \int \int f(X(u.v), Y(u,v)) J(u,v) du dv %}
when dealing with single variable integration, this formula reduces to
{% \int f(x)dx = \int f[g(t)]g'(t) dt %}
