Chapter 14.3, Problem 80E

To determine

(a)

Toverify:

for $\left(x,y\right)\ne \left(0,0\right)$ :

$f_x\left(x,y\right)=\frac{y\left(x^4+4x^2{y}^2-y^4\right)}{{\left(x^2+y^2\right)}^2}$, $f_y\left(x,y\right)=\frac{x\left(x^4-4x^2{y}^2-y^4\right)}{{\left(x^2+y^2\right)}^2}$

To determine

(b)

Toshow:

that $f_x\left(0,0\right)=f_y\left(0,0\right)=0$ and that $f_{yx}\left(0,0\right)$ and $f_{xy}\left(0,0\right)$ both exist but are not equal by using the limit definition of the partial derivative to show.

To determine

(c)

Toshow:

$f_{xy}\left(x,y\right)=f_{yx}\left(x,y\right)=\frac{x^6+9x^4{y}^2-9x^2{y}^4-y^6}{{\left(x^2+y^2\right)}^3}for\left(x,y\right)\ne \left(0,0\right)$ and $f_{xy}\left(x,y\right)$ is not continuous at $\left(0,0\right)$

To determine

(d)

Toexplain:

why the result of part (b) does not contradict Clairaut’s theorem.