Jacobian Determinants Examples 1

# Jacobian Determinants Examples 1

Recall from the Jacobian Determinants page that if $F(x, y, ...)$ and $G(x, y, ...)$ are functions, then the Jacobian Determinant of $F$ and $G$ with respect to $x$ and $y$ is the determinant:

(1)
\begin{align} \quad \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} \end{align}

Furthermore, if $F(x, y, z, ...)$, $G(x, y, z, ...)$ and $H(x, y, z, ...)$ are functions, then the Jacobian Determinant of $F$, $G$ and $H$ with respect to $x$, $y$, and $z$ is the determinant:

(2)
\begin{align} \quad \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} \end{align}

We will now look at some examples of computing Jacobian Determinants.

## Example 1

Let $F(x, y) = 2x^2 + 3 \sin y$ and let $G(x, y) = e^x - 2y$. Compute the Jacobian Determinant $\frac{\partial (F, G)}{\partial (x, y)}$.

We have that $\frac{\partial F}{\partial x} = 4x$, $\frac{\partial F}{\partial y} = 3 \cos y$, $\frac{\partial G}{\partial x} = e^x$ and $\frac{\partial G}{\partial y} = -2$. Therefore the Jacobian Determinant is:

(3)
\begin{align} \quad \frac{\partial (F, G)}{\partial (x, y)} = \begin{vmatrix} 4x & 3 \cos y \\ e^x & -2 \end{vmatrix} = -8x - 3e^x \cos y \end{align}