Jacobian Calculator

The calculator will find the Jacobian matrix of the set of functions and the Jacobian determinant (if possible), with steps shown.

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Calculate the Jacobian of $$$\left\{x = r \cos{\left(\theta \right)}, y = r \sin{\left(\theta \right)}\right\}$$$.

Solution

The Jacobian matrix is defined as follows: $$$J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta}\\\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta}\end{array}\right].$$$

In our case, $$$J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}\frac{\partial}{\partial r} \left(r \cos{\left(\theta \right)}\right) & \frac{\partial}{\partial \theta} \left(r \cos{\left(\theta \right)}\right)\\\frac{\partial}{\partial r} \left(r \sin{\left(\theta \right)}\right) & \frac{\partial}{\partial \theta} \left(r \sin{\left(\theta \right)}\right)\end{array}\right].$$$

Find the derivatives (for steps, see derivative calculator): $$$J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}\cos{\left(\theta \right)} & - r \sin{\left(\theta \right)}\\\sin{\left(\theta \right)} & r \cos{\left(\theta \right)}\end{array}\right].$$$

The Jacobian determinant is the determinant of the Jacobian matrix: $$$\left|\begin{array}{cc}\cos{\left(\theta \right)} & - r \sin{\left(\theta \right)}\\\sin{\left(\theta \right)} & r \cos{\left(\theta \right)}\end{array}\right| = r$$$ (for steps, see determinant calculator).

Answer

The Jacobian matrix is $$$\left[\begin{array}{cc}\cos{\left(\theta \right)} & - r \sin{\left(\theta \right)}\\\sin{\left(\theta \right)} & r \cos{\left(\theta \right)}\end{array}\right]$$$A.

The Jacobian determinant is $$$r$$$A.