Answer to Question #195218 in Calculus for Ichigo

Question #195218

Check the local inevitability of the function f defined by f(x,y)=(x²-y²,2xy) at (1,-1). Find a domain for the function f in which f is invertible. 


1
Expert's answer
2021-05-24T11:23:22-0400

Given, "f(x,y)=(x^2-y^2,2xy)"


Here, "u=x^2-y^2,v=2xy"


"J|f(x,y)|=\\begin{vmatrix} \\dfrac{du}{dx} & \\dfrac{du}{dy}\\\\\\\\\\dfrac{dv}{dx}&\\dfrac{dv}{dy}\\end{vmatrix}\n\n =\\begin{vmatrix} 2x& -2y\\\\2y&2x\\end{vmatrix}=4x^2+4y^2"


"\\Rightarrow J|f(1,-1)|=4(1)^2+4(-1)^2=4+4=8"


"Hence, J|f(1,-1)|\\neq 0," So F is invertible at (1,-1).


As "4(x^2+y^2)>0," So The domain for which f is invertible is all Real numbers R.



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