$$$\left\{x = 2 u \cos{\left(5 v \right)}, y = 2 \sin{\left(5 v \right)}\right\}$$$ 的雅可比矩陣及其行列式
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計算$$$\left\{x = 2 u \cos{\left(5 v \right)}, y = 2 \sin{\left(5 v \right)}\right\}$$$的雅可比矩陣。
解答
雅可比矩陣定義如下：$$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{array}\right]$$$。
在本例中，$$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}\frac{\partial}{\partial u} \left(2 u \cos{\left(5 v \right)}\right) & \frac{\partial}{\partial v} \left(2 u \cos{\left(5 v \right)}\right)\\\frac{\partial}{\partial u} \left(2 \sin{\left(5 v \right)}\right) & \frac{\partial}{\partial v} \left(2 \sin{\left(5 v \right)}\right)\end{array}\right]$$$。
求導數（步驟見 導數計算器）：$$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}2 \cos{\left(5 v \right)} & - 10 u \sin{\left(5 v \right)}\\0 & 10 \cos{\left(5 v \right)}\end{array}\right]$$$
雅可比行列式是雅可比矩陣的行列式：$$$\left|\begin{array}{cc}2 \cos{\left(5 v \right)} & - 10 u \sin{\left(5 v \right)}\\0 & 10 \cos{\left(5 v \right)}\end{array}\right| = 20 \cos^{2}{\left(5 v \right)}$$$（計算步驟請參見 行列式計算器）。
答案
雅可比矩陣為 $$$\left[\begin{array}{cc}2 \cos{\left(5 v \right)} & - 10 u \sin{\left(5 v \right)}\\0 & 10 \cos{\left(5 v \right)}\end{array}\right]$$$A。
雅可比行列式為 $$$20 \cos^{2}{\left(5 v \right)}$$$A。