Jacobi-matrisen och dess determinant för $$$\left\{x = 2 u \cos{\left(5 v \right)}, y = 2 \sin{\left(5 v \right)}\right\}$$$

Beräkna Jacobianen för $$$\left\{x = 2 u \cos{\left(5 v \right)}, y = 2 \sin{\left(5 v \right)}\right\}$$$.

Jacobianmatrisen definieras enligt följande: $$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{array}\right].$$$

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

Bestäm derivatorna (för steg, se derivataräknare): $$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}2 \cos{\left(5 v \right)} & - 10 u \sin{\left(5 v \right)}\\0 & 10 \cos{\left(5 v \right)}\end{array}\right]$$$

Jakobideterminanten är determinanten av Jakobimatrisen: $$$\left|\begin{array}{cc}2 \cos{\left(5 v \right)} & - 10 u \sin{\left(5 v \right)}\\0 & 10 \cos{\left(5 v \right)}\end{array}\right| = 20 \cos^{2}{\left(5 v \right)}$$$ (för steg, se determinantkalkylator).

Jacobi-matrisen är $$$\left[\begin{array}{cc}2 \cos{\left(5 v \right)} & - 10 u \sin{\left(5 v \right)}\\0 & 10 \cos{\left(5 v \right)}\end{array}\right]$$$A.

Jacobi-determinanten är $$$20 \cos^{2}{\left(5 v \right)}$$$A.