Matriz jacobiana de $$$\left\{x = 2 u \cos{\left(5 v \right)}, y = 2 \sin{\left(5 v \right)}\right\}$$$ y su determinante
Tu entrada
Calcule la matriz jacobiana de $$$\left\{x = 2 u \cos{\left(5 v \right)}, y = 2 \sin{\left(5 v \right)}\right\}$$$.
Solución
La matriz jacobiana se define de la siguiente manera: $$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{array}\right].$$$
En nuestro caso, $$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}\frac{\partial}{\partial u} \left(2 u \cos{\left(5 v \right)}\right) & \frac{\partial}{\partial v} \left(2 u \cos{\left(5 v \right)}\right)\\\frac{\partial}{\partial u} \left(2 \sin{\left(5 v \right)}\right) & \frac{\partial}{\partial v} \left(2 \sin{\left(5 v \right)}\right)\end{array}\right].$$$
Encuentre las derivadas (para los pasos, vea calculadora de derivadas): $$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}2 \cos{\left(5 v \right)} & - 10 u \sin{\left(5 v \right)}\\0 & 10 \cos{\left(5 v \right)}\end{array}\right].$$$
El determinante jacobiano es el determinante de la matriz jacobiana: $$$\left|\begin{array}{cc}2 \cos{\left(5 v \right)} & - 10 u \sin{\left(5 v \right)}\\0 & 10 \cos{\left(5 v \right)}\end{array}\right| = 20 \cos^{2}{\left(5 v \right)}$$$ (para los pasos, consulte la calculadora de determinantes).
Respuesta
La matriz jacobiana es $$$\left[\begin{array}{cc}2 \cos{\left(5 v \right)} & - 10 u \sin{\left(5 v \right)}\\0 & 10 \cos{\left(5 v \right)}\end{array}\right]$$$A.
El determinante jacobiano es $$$20 \cos^{2}{\left(5 v \right)}$$$A.