$$$\left\{u = x, v = y, w = x y\right\}$$$ 的雅可比矩阵及其行列式
您的输入
计算$$$\left\{u = x, v = y, w = x y\right\}$$$的雅可比矩阵。
解答
雅可比矩阵定义如下：$$$J{\left(u,v,w \right)}\left(x, y\right) = \left[\begin{array}{cc}\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\\\frac{\partial w}{\partial x} & \frac{\partial w}{\partial y}\end{array}\right]$$$。
在我们的情况下，$$$J{\left(u,v,w \right)}\left(x, y\right) = \left[\begin{array}{cc}\frac{\partial}{\partial x} \left(x\right) & \frac{\partial x}{\partial y}\\\frac{\partial y}{\partial x} & \frac{\partial}{\partial y} \left(y\right)\\\frac{\partial}{\partial x} \left(x y\right) & \frac{\partial}{\partial y} \left(x y\right)\end{array}\right]$$$。
求导数（步骤详见导数计算器）：$$$J{\left(u,v,w \right)}\left(x, y\right) = \left[\begin{array}{cc}1 & 0\\0 & 1\\y & x\end{array}\right]$$$。
由于矩阵不是方阵，雅可比行列式不存在。
答案
雅可比矩阵为 $$$\left[\begin{array}{cc}1 & 0\\0 & 1\\y & x\end{array}\right]$$$A。
雅可比行列式不存在。