Jacobian and its determinant of $$$\left\{x = 2 u \cos{\left(5 v \right)}, y = 2 \sin{\left(5 v \right)}\right\}$$$

The calculator will find the Jacobian (and its determinant) of the set of the functions (or the transformation) $$$\left\{x = 2 u \cos{\left(5 v \right)}, y = 2 \sin{\left(5 v \right)}\right\}$$$, with steps shown.
Leave empty for autodetection or specify variables like x,y (comma-separated).

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Your Input

Calculate the Jacobian of $$$\left\{x = 2 u \cos{\left(5 v \right)}, y = 2 \sin{\left(5 v \right)}\right\}$$$.


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

In our case, $$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}\frac{\partial}{\partial u} \left(2 u \cos{\left(5 v \right)}\right) & \frac{\partial}{\partial v} \left(2 u \cos{\left(5 v \right)}\right)\\\frac{\partial}{\partial u} \left(2 \sin{\left(5 v \right)}\right) & \frac{\partial}{\partial v} \left(2 \sin{\left(5 v \right)}\right)\end{array}\right].$$$

Find the derivatives (for steps, see derivative calculator): $$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}2 \cos{\left(5 v \right)} & - 10 u \sin{\left(5 v \right)}\\0 & 10 \cos{\left(5 v \right)}\end{array}\right].$$$

The Jacobian determinant is the determinant of the Jacobian matrix: $$$\left|\begin{array}{cc}2 \cos{\left(5 v \right)} & - 10 u \sin{\left(5 v \right)}\\0 & 10 \cos{\left(5 v \right)}\end{array}\right| = 20 \cos^{2}{\left(5 v \right)}$$$ (for steps, see determinant calculator).


The Jacobian matrix is $$$\left[\begin{array}{cc}2 \cos{\left(5 v \right)} & - 10 u \sin{\left(5 v \right)}\\0 & 10 \cos{\left(5 v \right)}\end{array}\right]$$$A.

The Jacobian determinant is $$$20 \cos^{2}{\left(5 v \right)}$$$A.