How to Find Jacobian of functions given below

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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

\[\left(\begin{matrix}\frac{\partial f}{\partial x}&\frac{\partial f}{\partial y}\\\frac{\partial g}{\partial x}&\frac{\partial g}{\partial y}\\\end{matrix}\right)\] \[\left(\begin{matrix}-2xy&-2y^2\\-2x^2&-2xy\\\end{matrix}\right)\]


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