Question : Find Jacobian of functions given below
\[f(x,y)=y\left(1-\left(x^2+y^2\right)\right) , g(x,y)=x\left(1-\left(x_2+y^2\right)\right) \]
Solution :
given functions
\[f(x,y)=y\left(1-\left(x^2+y^2\right)\right) , g(x,y)=x\left(1-\left(x_2+y^2\right)\right) \]
Their first order partial derivatives are
\frac{\partial f}{\partial x} = -2xy\]
\[\frac{\partial f}{\partial y} = -2y^2\]
\[\frac{\partial g}{\partial x} = -2x^2\]
\[\frac{\partial g}{\partial y} =-2xy\]
Then the Jacobian of the functions f and g is given as
\[\left(\begin{matrix}\frac{\partial f}{\partial x}&\frac{\partial f}{\partial y}\\\frac{\partial g}{\partial x}&\frac{\partial g}{\partial y}\\\end{matrix}\right)\] \[\left(\begin{matrix}-2xy&-2y^2\\-2x^2&-2xy\\\end{matrix}\right)\]