$$$\left\{x = 3 e^{- 4 r} \sin{\left(3 \theta \right)}, y = e^{4 r} \cos{\left(3 \theta \right)}\right\}$$$ 的雅可比矩陣及其行列式
您的輸入
計算$$$\left\{x = 3 e^{- 4 r} \sin{\left(3 \theta \right)}, y = e^{4 r} \cos{\left(3 \theta \right)}\right\}$$$的雅可比矩陣。
解答
雅可比矩陣定義如下：$$$J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta}\\\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta}\end{array}\right]$$$。
在本例中，$$$J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}\frac{\partial}{\partial r} \left(3 e^{- 4 r} \sin{\left(3 \theta \right)}\right) & \frac{\partial}{\partial \theta} \left(3 e^{- 4 r} \sin{\left(3 \theta \right)}\right)\\\frac{\partial}{\partial r} \left(e^{4 r} \cos{\left(3 \theta \right)}\right) & \frac{\partial}{\partial \theta} \left(e^{4 r} \cos{\left(3 \theta \right)}\right)\end{array}\right]$$$。
求導數（步驟見 導數計算器）：$$$J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}- 12 e^{- 4 r} \sin{\left(3 \theta \right)} & 9 e^{- 4 r} \cos{\left(3 \theta \right)}\\4 e^{4 r} \cos{\left(3 \theta \right)} & - 3 e^{4 r} \sin{\left(3 \theta \right)}\end{array}\right]$$$
雅可比行列式是雅可比矩陣的行列式：$$$\left|\begin{array}{cc}- 12 e^{- 4 r} \sin{\left(3 \theta \right)} & 9 e^{- 4 r} \cos{\left(3 \theta \right)}\\4 e^{4 r} \cos{\left(3 \theta \right)} & - 3 e^{4 r} \sin{\left(3 \theta \right)}\end{array}\right| = - 36 \cos{\left(6 \theta \right)}$$$（計算步驟請參見 行列式計算器）。
答案
雅可比矩陣為 $$$\left[\begin{array}{cc}- 12 e^{- 4 r} \sin{\left(3 \theta \right)} & 9 e^{- 4 r} \cos{\left(3 \theta \right)}\\4 e^{4 r} \cos{\left(3 \theta \right)} & - 3 e^{4 r} \sin{\left(3 \theta \right)}\end{array}\right]$$$A。
雅可比行列式為 $$$- 36 \cos{\left(6 \theta \right)}$$$A。