Question

Does the function

 f(x,y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}},x\neq 0,y\neq 0

satisfy the requirement of Schwarz’s theorem at ?)1,1( Justify your answer.

07 Feb 2021
Answer :
Word Count : 550
To solve this problem, we need to determine if the function \( f(x, y) = \frac{x^2 - y^2}{x^2 + y^2} \) satisfies the conditions of Schwarz's theorem at the point \( (1, 1) \). ### Schwarz's Theorem Schwarz's theorem states that if a function \( f(x, y) \) is continuously differentiable in a neighborhood of a point and if the mixed partial derivatives are equal (i.e., \( \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} \)), then Schwarz’s theorem holds. ### Steps to Solve 1. Find the partial ______ _______ __________ ___ _________ ______ _____ __________ ______.
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