Jacobiano e o seu determinante de $$$\left\{x = 3 e^{- 4 r} \sin{\left(3 \theta \right)}, y = e^{4 r} \cos{\left(3 \theta \right)}\right\}$$$

Calcule a matriz jacobiana de $$$\left\{x = 3 e^{- 4 r} \sin{\left(3 \theta \right)}, y = e^{4 r} \cos{\left(3 \theta \right)}\right\}$$$.

A matriz jacobiana é definida da seguinte forma: $$$J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta}\\\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta}\end{array}\right].$$$

No nosso caso, $$$J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}\frac{\partial}{\partial r} \left(3 e^{- 4 r} \sin{\left(3 \theta \right)}\right) & \frac{\partial}{\partial \theta} \left(3 e^{- 4 r} \sin{\left(3 \theta \right)}\right)\\\frac{\partial}{\partial r} \left(e^{4 r} \cos{\left(3 \theta \right)}\right) & \frac{\partial}{\partial \theta} \left(e^{4 r} \cos{\left(3 \theta \right)}\right)\end{array}\right].$$$

Encontre as derivadas (para os passos, veja calculadora de derivadas): $$$J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}- 12 e^{- 4 r} \sin{\left(3 \theta \right)} & 9 e^{- 4 r} \cos{\left(3 \theta \right)}\\4 e^{4 r} \cos{\left(3 \theta \right)} & - 3 e^{4 r} \sin{\left(3 \theta \right)}\end{array}\right]$$$

O determinante jacobiano é o determinante da matriz jacobiana: $$$\left|\begin{array}{cc}- 12 e^{- 4 r} \sin{\left(3 \theta \right)} & 9 e^{- 4 r} \cos{\left(3 \theta \right)}\\4 e^{4 r} \cos{\left(3 \theta \right)} & - 3 e^{4 r} \sin{\left(3 \theta \right)}\end{array}\right| = - 36 \cos{\left(6 \theta \right)}$$$ (para ver as etapas, consulte calculadora de determinante).

A matriz Jacobiana é $$$\left[\begin{array}{cc}- 12 e^{- 4 r} \sin{\left(3 \theta \right)} & 9 e^{- 4 r} \cos{\left(3 \theta \right)}\\4 e^{4 r} \cos{\left(3 \theta \right)} & - 3 e^{4 r} \sin{\left(3 \theta \right)}\end{array}\right]$$$A.

O determinante jacobiano é $$$- 36 \cos{\left(6 \theta \right)}$$$A.