Word Count : 609

---

We are asked whether the function $$ f(x,y) = \frac{x^2 - y^2}{x^2 + y^2}, \quad x \neq 0, y \neq 0 $$ satisfies Schwarz’s Theorem (equality of mixed partial derivatives) at the point $(1,1)$. --- ### Step 1: Recall Schwarz’s Theorem Schwarz’s (Clairaut’s) theorem states: > If $f_{xy}$ and $f_{yx}$ are continuous in a neighborhood of $(a,b)$, then > > $$ > f_{xy}(a,b) = f_{yx}(a,b) > $$ So we need: 1. $f_x$, $f_y$ to exist. 2. $f_{xy}$ and $f_{yx}$ to exist and be continuous at $(1,1)$. --- ### Step 2: Compute First Partial Derivatives Let $$ f(x,y) = \frac{x^2 - y^2}{x^2 + y^2} $$ We use the quotient rule: --- (a) Partial derivative with respect to $x$ $$ f_x = \frac{(2x)(x^2+y^2) - (x^2-y^2)(2x)}{(x^2+y^2)^2} $$ Simplify numerator: $$ 2x(x^2+y^2) - 2x(x^2-y^2) $$ $$ = 2x \big[ (x^2+y^2) - (x^2-y^2) \big] $$ $$ = 2x (2y^2) $$ $$ f_x = \frac{4xy^2}{(x^2+y^2)^2} $$ --- (b) ___ __________ __________ _________ _________ __________ ___ ___ ________ ________ _____.
________ _______ ______ ______ ________ _____ ___ _________ ______.
______ ___ ___ ___ ________ _______ ______ _____.
________ ___ __________ ________ __________ _____ _________ _________ ___ ____ _____.
____ _______ ______ __________ ___ _____ ____ _______.
___ __________ __________ ________ __________ ______ _____.
___ __________ _______ _____ ____ _______ _____.
___ ________ __________ ____ ______ __________.
__________ ___ _____ ________ ___ _________ ______.
________ ______ ________ _________ _______ _____ _____ ______ _________ _________ _________.
___ ______ ___ ____ ________ ____.
_______ __________ __________ ________ ____ _____ _________ ______.
______ ____ ___ _________ __________ ___ ______ ________ ______.
________ _______ ______ __________ ______ ________ ___ __________ _________ ___.
______ ________ ________ ________ _________ _______.
__________ ________ _________ _____ ________ ________ _________ ___ _______ __________.
_____ _________ __________ ______ ________ _______.
__________ _________ __________ ___ ____ ___ ____ ___ __________.
____ ___ __________ ___ _____ ______ ____ _____ _________ _______ _______.
______ _____ ____ __________ __________ __________ ____ ________ ___ _________ _____ __________.
_________ __________ _______ ______ _______.
_________ _____ ___ ____ ____ ___.
_____ ___ ______ ________ __________ ________ __________ ____ ____ _______.
_______ ________ ____ _________ ________ _______ ___ ____ ________ _________ ________ __________.
________ __________ ________ _______ _______ ________ _________ _____.
_______ ______ ____ _____ _________ ___.
______ __________ ____ __________ _______.
______ __________ ________ ___ ____ _____ ________ _______ _________ ____.
_______ ______ _______ _________ __________.
____ ____ ___ _____ __________ ________ ________ _____.
_______ ___ _________ ________ ________ __________ ______ ______ _________ ____ _________.
___ __________ ____ ___ _______ _______ ___ __________.
______ _____ ___ ____ ______ ____ _______ __________ __________ ____ __________ ___.
______ _______ ___ ___ ______.
________ ___ ________ _________ _____ ____ ___ __________ __________ _______.
____ ______ ______ _____ _____ _________ ___ ____.
________ ______ _____ _______ _____ _____ ______.
____ _________ ___ ____ ______ _______ _____ ____ __________ _____ ______.
_____ _______ _________ _________ ____ ____ _____.
______ _______ __________ __________ ___ _____ ______ _____.
_________ ______ ___ ________ __________ ____ _____ ____ ___ ____ __________.
____ _______ _________ __________ ______ ______ _____ ________.
_____ __________ ___ _______ __________ __________ __________.
____ _____ __________ _____ ______.
____ _________ ___ _______ _____ ___ ______.
______ ______ ________ _________ ______ _____ ____ _____ ______ ________.
____ __________ __________ ___ ________ ___ _______ ________ ____ __________ __________ ___.
______ ______ __________ _________ ______ ___ _________ __________ ____ _______ ______.
__________ ________ __________ ________ ___.
____ _______ ___ __________ ______.
______ _______ __________ __________ __________ ________ ________ ________ _________ ___.
_____ ____ __________ _____ _________.
________ _______ _________ _________ ________ ______ _____ ________ _____.
______ ______ ___ ____ __________ ____ _________ ______ _____ ________.
__________ _______ __________ _____ _______ __________.
_________ ______.
Get Full Answer on WhatsApp