$x^2 u_x\+y^2 u_y\+z^2 u_z$

$\=x^2 \left(F_r r_x\+F_s s_x\right)\+y^2 \left(F_r r_y\+F_s s_y\right)\+z^2 \left(F_r r_z\+F_s s_z\right)$

$\=x^2 \left(F_r \left(-\frac{1}{x^2}\right)\+F_s \left(-\frac{1}{x^2}\right)\right)\+y^2 \left(F_r \left(\frac{1}{y^2}\right)\+F_s \left(0\right)\right)\+z^2 \left(F_r \left(0\right)\+F_s \left(\frac{1}{z^2}\right)\right)$

$\=\left(-F_r-F_s\right)\+\left(F_r\+0\right)\+\left(0\+F_s\right)$

$\=\left(-1\+1\right)F_r\+\left(-1\+1\right)F_s$

$\=0\cdot F_r\+0\cdot F_s$

$\=0\+0$

$\=0$

Define $r\,s$, and $u$ 

$r\=\frac{y-x}{x y}$$\stackrel{\text{assign}}{\to }$

$s\=\frac{z-x}{x z}$$\stackrel{\text{assign}}{\to }$

$u\=F\left(r\,s\right)$$\stackrel{\text{assign}}{\to }$

Compute $x^2 u_x\+y^2 u_y\+z^2 u_z$ and simplify the result to 0

$x^2 \frac{\partial }{\partial x} u\+y^2 \frac{\partial }{\partial y} u\+z^2 \frac{\partial }{\partial z} u$ = $x^2\left({\mathrm{D}}_1\left(F\right)\left(\frac{y-x}{xy}\,\frac{z-x}{xz}\right)\left(-\frac{1}{xy}-\frac{y-x}{x^{2}y}\right)\+{\mathrm{D}}_2\left(F\right)\left(\frac{y-x}{xy}\,\frac{z-x}{xz}\right)\left(-\frac{1}{xz}-\frac{z-x}{x^{2}z}\right)\right)\+y^2{\mathrm{D}}_1\left(F\right)\left(\frac{y-x}{xy}\,\frac{z-x}{xz}\right)\left(\frac{1}{xy}-\frac{y-x}{x{y}^2}\right)\+z^2{\mathrm{D}}_2\left(F\right)\left(\frac{y-x}{xy}\,\frac{z-x}{xz}\right)\left(\frac{1}{xz}-\frac{z-x}{x{z}^2}\right)$$\stackrel{\text{simplify}}{\=}$$0$$$

Separately obtain and simplify the partial derivatives $u_x\,u_y$, and $u_z$ 

$\frac{\partial }{\partial x} u$ = ${\mathrm{D}}_1\left(F\right)\left(\frac{y-x}{xy}\,\frac{z-x}{xz}\right)\left(-\frac{1}{xy}-\frac{y-x}{x^{2}y}\right)\+{\mathrm{D}}_2\left(F\right)\left(\frac{y-x}{xy}\,\frac{z-x}{xz}\right)\left(-\frac{1}{xz}-\frac{z-x}{x^{2}z}\right)$$\stackrel{\text{simplify}}{\=}$$-\frac{{\mathrm{D}}_1\left(F\right)\left(-\frac{-y\+x}{xy}\,-\frac{-z\+x}{xz}\right)\+{\mathrm{D}}_2\left(F\right)\left(-\frac{-y\+x}{xy}\,-\frac{-z\+x}{xz}\right)}{x^2}$$$

$\frac{\partial }{\partial y} u$ = ${\mathrm{D}}_1\left(F\right)\left(\frac{y-x}{xy}\,\frac{z-x}{xz}\right)\left(\frac{1}{xy}-\frac{y-x}{x{y}^2}\right)$$\stackrel{\text{simplify}}{\=}$$\frac{{\mathrm{D}}_1\left(F\right)\left(-\frac{-y\+x}{xy}\,-\frac{-z\+x}{xz}\right)}{y^2}$$$

$\frac{\partial }{\partial z} u$ = ${\mathrm{D}}_2\left(F\right)\left(\frac{y-x}{xy}\,\frac{z-x}{xz}\right)\left(\frac{1}{xz}-\frac{z-x}{x{z}^2}\right)$$\stackrel{\text{simplify}}{\=}$$\frac{{\mathrm{D}}_2\left(F\right)\left(-\frac{-y\+x}{xy}\,-\frac{-z\+x}{xz}\right)}{z^2}$$$

Obtain and simplify $r_x\,r_y\,r_z\,s_x\,s_y$, and $s_z$ 

$\frac{\partial }{\partial x} r$ = $-\frac{1}{xy}-\frac{y-x}{x^{2}y}$$\stackrel{\text{simplify}}{\=}$$-\frac{1}{x^2}$$$

$\frac{\partial }{\partial x} s$ = $-\frac{1}{xz}-\frac{z-x}{x^{2}z}$$\stackrel{\text{simplify}}{\=}$$-\frac{1}{x^2}$$$

$\frac{\partial }{\partial y} r$ = $\frac{1}{xy}-\frac{y-x}{x{y}^2}$$\stackrel{\text{simplify}}{\=}$$\frac{1}{y^2}$$$

$\frac{\partial }{\partial y} s$ = $0$$$

$\frac{\partial }{\partial z} r$ = $0$$$

$\frac{\partial }{\partial z} s$ = $\frac{1}{xz}-\frac{z-x}{x{z}^2}$$\stackrel{\text{simplify}}{\=}$$\frac{1}{z^2}$$$

Initialize

$r≔\frac{y-x}{x y}\:$

$s≔\frac{z-x}{x z}\:$

$u≔F\left(r\,s\right)\:$

Apply the diff and simplify commands to evaluate the expression $x^2 u_x\+y^2 u_y\+z^2 u_z$ 

$\mathrm{simplify}\left(x^2 \mathrm{diff}\left(u\,x\right)\+y^2 \mathrm{diff}\left(u\,y\right)\+z^2 \mathrm{diff}\left(u\,z\right)\right)$ = $0$$$

Separately obtain and simplify $u_x\,u_y$, and $u_z$ 

$\mathrm{simplify}\left(\mathrm{diff}\left(u\,x\right)\right)$ = $-\frac{{\mathrm{D}}_1\left(F\right)\left(-\frac{-y+x}{xy}\,-\frac{-z+x}{xz}\right)+{\mathrm{D}}_2\left(F\right)\left(-\frac{-y+x}{xy}\,-\frac{-z+x}{xz}\right)}{x^2}$$$

$\mathrm{simplify}\left(\mathrm{diff}\left(u\,y\right)\right)$ = $\frac{{\mathrm{D}}_1\left(F\right)\left(-\frac{-y+x}{xy}\,-\frac{-z+x}{xz}\right)}{y^2}$$$

$\mathrm{simplify}\left(\mathrm{diff}\left(u\,z\right)\right)$ = $\frac{{\mathrm{D}}_2\left(F\right)\left(-\frac{-y+x}{xy}\,-\frac{-z+x}{xz}\right)}{z^2}$$$

Obtain and simplify the derivatives $r_x\,r_y\,r_z$ and $s_x\,s_y\,s_z$ 

$\mathrm{simplify}\left(\mathrm{diff}\left(r\,x\right)\right)$ = $-\frac{1}{x^2}$$$

$\mathrm{simplify}\left(\mathrm{diff}\left(s\,x\right)\right)$ = $-\frac{1}{x^2}$$$

$\mathrm{simplify}\left(\mathrm{diff}\left(r\,y\right)\right)$ = $\frac{1}{y^2}$$$

$\mathrm{simplify}\left(\mathrm{diff}\left(s\,y\right)\right)$ = $0$$$

$\mathrm{simplify}\left(\mathrm{diff}\left(r\,z\right)\right)$ = $0$$$

$\mathrm{simplify}\left(\mathrm{diff}\left(s\,z\right)\right)$ = $\frac{1}{z^2}$$$