$$$\left\{u = x, v = y, w = x y\right\}$$$ 的雅可比矩陣及其行列式
您的輸入
計算$$$\left\{u = x, v = y, w = x y\right\}$$$的雅可比矩陣。
解答
雅可比矩陣定義如下：$$$J{\left(u,v,w \right)}\left(x, y\right) = \left[\begin{array}{cc}\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\\\frac{\partial w}{\partial x} & \frac{\partial w}{\partial y}\end{array}\right]$$$。
在本例中，$$$J{\left(u,v,w \right)}\left(x, y\right) = \left[\begin{array}{cc}\frac{\partial}{\partial x} \left(x\right) & \frac{\partial x}{\partial y}\\\frac{\partial y}{\partial x} & \frac{\partial}{\partial y} \left(y\right)\\\frac{\partial}{\partial x} \left(x y\right) & \frac{\partial}{\partial y} \left(x y\right)\end{array}\right]$$$。
求導數（步驟見 導數計算器）：$$$J{\left(u,v,w \right)}\left(x, y\right) = \left[\begin{array}{cc}1 & 0\\0 & 1\\y & x\end{array}\right]$$$
由於矩陣不是方陣，雅可比行列式不存在。
答案
雅可比矩陣為 $$$\left[\begin{array}{cc}1 & 0\\0 & 1\\y & x\end{array}\right]$$$A。
雅可比行列式不存在。