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

### Solution

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.