Jacobian Determinants

# Jacobian Determinants

Recall from the Systems of Multivariable Equations that we can often times find partial derivatives at solutions to multivariable systems of equations. Recall that if we have the following system of two multivariable equations $F$ and $G$ (all of whose first partial derivatives are continuous) in the four variables $x$, $y$, $z$, and $w$:

(1)
\begin{align} \left\{\begin{matrix} F(x, y, z, w) = 0\\ G(x, y, z, w) = 0 \end{matrix}\right. \end{align}

Then we can find various partial derivatives to this system. We already saw that $\left ( \frac{\partial x}{\partial z} \right )_w$ (the partial derivative of $x$ with respect to $z$ holding $w$ as a fixed independent variable) could be computed with the following formula:

(2)
\begin{align} \left ( \frac{\partial x}{\partial z} \right )_w = \frac{\det \begin{vmatrix} -F_3 & F_2\\ -G_3 & G_2 \end{vmatrix}}{\det \begin{vmatrix} F_1 & F_2\\ G_1 & G_2 \end{vmatrix}} = \frac{-F_3G_2 + F_2G_3}{F_1G_2 - F_2G_1} \end{align}

Such determinants that appear in similar formulas are important and have a special name which we define below.

 Definition: The Jacobian Determinant of the two functions $F(x, y, ...)$ and $G(x, y, ...)$ with respect to the variables $x$ and $y$ denoted $\frac{\partial (F, G)}{\partial (x, y)} = \begin{vmatrix} \frac{\partial F}{\partial x} & \frac{\partial F}{\partial y}\\ \frac{\partial G}{\partial x} & \frac{\partial G}{\partial y} \end{vmatrix} = \begin{vmatrix} F_1 & F_2\\ G_1 & G_2 \end{vmatrix}$. The Jacobian Determinant of the three functions $F(x, y, z, ...)$, $G(x, y, z, ...)$ and $H(x, y, z, ...)$ with respect to the variables $x$, $y$, and $z$ denoted $\frac{\partial (F, G, H)}{\partial (x, y, z)} = \begin{vmatrix} \frac{\partial F}{\partial x} & \frac{\partial F}{\partial y} & \frac{\partial F}{\partial z}\\ \frac{\partial G}{\partial x} & \frac{\partial G}{\partial y} & \frac{\partial G}{\partial z}\\ \frac{\partial H}{\partial x} & \frac{\partial H}{\partial y} & \frac{\partial H}{\partial z} \end{vmatrix} = \begin{vmatrix} F_1 & F_2 & F_3\\ G_1 & G_2 & G_3\\ H_1 & H_2 & H_3\end{vmatrix}$.

The Jacobian Determinant for $n$ functions with respect to $n$ variables is defined analogously.

For the example above, we can rewrite the formula for $\left ( \frac{\partial x}{\partial z} \right )_w$ in terms of Jacobians:

(3)
\begin{align} \quad \left ( \frac{\partial x}{\partial z} \right )_w = - \frac{\frac{\partial (F, G)}{\partial (z, y)}}{\frac{\partial (F, G)}{\partial (x, y)}} \end{align}