$r_x\=\frac{x}{r}$

${\mathrm{\θ}}_x\=-\frac{y\/x^2}{1\+{\left(y\/x\right)}^2}\=-\frac{y}{x^2\+y^2}\=-\frac{y}{r^2}$

$r_y\=\frac{y}{r}$

${\mathrm{\θ}}_y\=\frac{1\/x}{1\+{\left(y\/x\right)}^2}\=\frac{x}{x^2\+y^2}\=\frac{x}{r^2}$

$r_{\mathrm{xx}}\=\frac{r-x\left(x\/r\right)}{r^2}\=\frac{r^2-x^2}{r^3}\=\frac{y^2}{r^3}$

${\mathrm{\θ}}_{\mathrm{xx}}\=2 y r^{-3}\left(x\/r\right)\=\frac{2 x y}{r^4}$

$r_{\mathrm{yy}}\=\frac{r-y\left(y\/r\right)}{r^2}\=\frac{r^2-y^2}{r^3}\=\frac{x^2}{r^3}$

${\mathrm{\θ}}_{\mathrm{yy}}\=-2 x r^{-3}\left(y\/r\right)\=-\frac{2 x y}{r^4}$

Table 9.3.15(a)   Derivatives of $r$ and $\mathrm{\θ}$ 

$f_x\=F_r r_x\+F_{\mathrm{\θ}} {\mathrm{\θ}}_x$

$f_y\=F_r r_y\+F_{\mathrm{\theta }} {\mathrm{\theta }}_y$

Table 9.3.15(b)   First partials by chain rule

$f_{\mathrm{xx}}\=\left(F_{\mathrm{rr}} r_x\+F_{r\mathrm{\θ}} {\mathrm{\θ}}_x\right) r_x\+f_r r_{\mathrm{xx}}\+\left(F_{\mathrm{\θ}r}{r}_x\+F_{\mathrm{\θ\θ}} {\mathrm{\θ}}_x\right) {\mathrm{\θ}}_x\+F_{\mathrm{\θ}} {\mathrm{\θ}}_{\mathrm{xx}}$

$f_{\mathrm{yy}}\=\left(F_{\mathrm{rr}} r_y\+F_{r\mathrm{\theta }} {\mathrm{\theta }}_y\right) r_y\+f_r r_{\mathrm{yy}}\+\left(F_{\mathrm{\theta }r}{r}_y\+F_{\mathrm{\theta \theta }} {\mathrm{\theta }}_y\right) {\mathrm{\theta }}_y\+F_{\mathrm{\theta }} {\mathrm{\theta }}_{\mathrm{yy}}$

Table 9.3.15(c)   Second partials by chain rule

$f_{\mathrm{xx}}\+f_{\mathrm{yy}}$

$\=F_{\mathrm{rr}} \left(r_x^2\+r_y^2\right)\+2 F_{r\mathrm{\θ}} \left(r_x {\mathrm{\θ}}_x\+r_y {\mathrm{\θ}}_y\right)\+F_{\mathrm{\θ\θ}} \left({\mathrm{\θ}}_x^2\+{\mathrm{\θ}}_y^2\right)\+F_r \left(R_{\mathrm{xx}}\+r_{\mathrm{yy}}\right)\+F_{\mathrm{\θ}} \left({\mathrm{\θ}}_{\mathrm{xx}}\+{\mathrm{\θ}}_{\mathrm{yy}}\right)$

$\=F_{\mathrm{rr}} \left(\frac{x^2}{r^2}\+\frac{y^2}{r^2}\right)\+2 F_{r\mathrm{\θ}} \left(-\frac{x y}{r^3}\+\frac{x y}{r^3}\right)\+F_{\mathrm{\θ\θ} }\left(\frac{y^2}{r^4}\+\frac{x^2}{r^4}\right)\+F_r \left(\frac{x^2}{r^3}\+\frac{y^2}{r^3}\right)\+F_{\mathrm{\θ}} \left(\frac{2 x y}{r^4}-\frac{2 x y}{r^4}\right)$

$\=F_{\mathrm{rr}} \left(\frac{r^2}{r^2}\right)\+2 F_{r\mathrm{\theta }} \left(0\right)\+F_{\mathrm{\theta \theta } }\left(\frac{r^2}{r^4}\right)\+F_r \left(\frac{r^2}{r^3}\right)\+F_{\mathrm{\theta }} \left(0\right)$

$\=F_{\mathrm{rr}}\+F_{\mathrm{\θ\θ}}\/r^2\+F_r\/r$

Define $f\left(x\,y\right)\=F\left(r\,\mathrm{\θ}\right)\=F\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)$

$F\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)$$\stackrel{\text{assign to a name}}{\to }$$f$

Obtain and simplify $f_{\mathrm{xx}}\+f_{\mathrm{yy}}$ 

$\frac{{\partial }^2}{{\partial }^{}{x}^2} f\+\frac{{\partial }^2}{{\partial }^{}{y}^2} f$

$\frac{\left(\frac{{\mathrm{D}}_{1\,1}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)x}{\sqrt{x^2\+y^2}}-\frac{{\mathrm{D}}_{1\,2}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)y}{x^2\left(1\+\frac{y^2}{x^2}\right)}\right)x}{\sqrt{x^2\+y^2}}-\frac{{\mathrm{D}}_1\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)x^2}{{\left(x^2\+y^2\right)}^{3\/2}}\+\frac{2{\mathrm{D}}_1\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)}{\sqrt{x^2\+y^2}}-\frac{\left(\frac{{\mathrm{D}}_{1\,2}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)x}{\sqrt{x^2\+y^2}}-\frac{{\mathrm{D}}_{2\,2}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)y}{x^2\left(1\+\frac{y^2}{x^2}\right)}\right)y}{x^2\left(1\+\frac{y^2}{x^2}\right)}\+\frac{2{\mathrm{D}}_2\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)y}{x^3\left(1\+\frac{y^2}{x^2}\right)}-\frac{2{\mathrm{D}}_2\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)y^3}{x^5{\left(1\+\frac{y^2}{x^2}\right)}^2}\+\frac{\left(\frac{{\mathrm{D}}_{1\,1}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)y}{\sqrt{x^2\+y^2}}\+\frac{{\mathrm{D}}_{1\,2}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)}{x\left(1\+\frac{y^2}{x^2}\right)}\right)y}{\sqrt{x^2\+y^2}}-\frac{{\mathrm{D}}_1\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)y^2}{{\left(x^2\+y^2\right)}^{3\/2}}\+\frac{\frac{{\mathrm{D}}_{1\,2}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)y}{\sqrt{x^2\+y^2}}\+\frac{{\mathrm{D}}_{2\,2}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)}{x\left(1\+\frac{y^2}{x^2}\right)}}{x\left(1\+\frac{y^2}{x^2}\right)}-\frac{2{\mathrm{D}}_2\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)y}{x^3{\left(1\+\frac{y^2}{x^2}\right)}^2}$

$\stackrel{\text{simplify}}{\=}$

$\frac{{\mathrm{D}}_{1\,1}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)x^2\sqrt{x^2\+y^2}\+{\mathrm{D}}_{1\,1}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}{y}^2\+{\mathrm{D}}_1\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)x^2\+{\mathrm{D}}_1\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)y^2\+{\mathrm{D}}_{2\,2}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}}{{\left(x^2\+y^2\right)}^{3\/2}}$

$\stackrel{\text{assign to a name}}{\to }$

$\mathrm{Temp}$

Replace Cartesian coordinates with polar coordinates and simplify

$\genfrac{}{}{0ex}{}{\mathrm{Temp}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=r \cos \left(\mathrm{\θ}\right)\,y\=r \sin \left(\mathrm{\θ}\right)}$

$\frac{{\mathrm{D}}_{1\,1}\left(F\right)\left(\sqrt{r^2{\cos \left(\mathrm{\θ}\right)}^2\+r^2{\sin \left(\mathrm{\θ}\right)}^2}\,\arctan \left(r\sin \left(\mathrm{\θ}\right)\,r\cos \left(\mathrm{\θ}\right)\right)\right)r^2{\cos \left(\mathrm{\θ}\right)}^2\sqrt{r^2{\cos \left(\mathrm{\θ}\right)}^2\+r^2{\sin \left(\mathrm{\θ}\right)}^2}\+{\mathrm{D}}_{1\,1}\left(F\right)\left(\sqrt{r^2{\cos \left(\mathrm{\θ}\right)}^2\+r^2{\sin \left(\mathrm{\θ}\right)}^2}\,\arctan \left(r\sin \left(\mathrm{\θ}\right)\,r\cos \left(\mathrm{\θ}\right)\right)\right)\sqrt{r^2{\cos \left(\mathrm{\θ}\right)}^2\+r^2{\sin \left(\mathrm{\θ}\right)}^2}{r}^2{\sin \left(\mathrm{\θ}\right)}^2\+{\mathrm{D}}_1\left(F\right)\left(\sqrt{r^2{\cos \left(\mathrm{\θ}\right)}^2\+r^2{\sin \left(\mathrm{\θ}\right)}^2}\,\arctan \left(r\sin \left(\mathrm{\θ}\right)\,r\cos \left(\mathrm{\θ}\right)\right)\right)r^2{\cos \left(\mathrm{\θ}\right)}^2\+{\mathrm{D}}_1\left(F\right)\left(\sqrt{r^2{\cos \left(\mathrm{\θ}\right)}^2\+r^2{\sin \left(\mathrm{\θ}\right)}^2}\,\arctan \left(r\sin \left(\mathrm{\θ}\right)\,r\cos \left(\mathrm{\θ}\right)\right)\right)r^2{\sin \left(\mathrm{\θ}\right)}^2\+{\mathrm{D}}_{2\,2}\left(F\right)\left(\sqrt{r^2{\cos \left(\mathrm{\θ}\right)}^2\+r^2{\sin \left(\mathrm{\θ}\right)}^2}\,\arctan \left(r\sin \left(\mathrm{\θ}\right)\,r\cos \left(\mathrm{\θ}\right)\right)\right)\sqrt{r^2{\cos \left(\mathrm{\θ}\right)}^2\+r^2{\sin \left(\mathrm{\θ}\right)}^2}}{{\left(r^2{\cos \left(\mathrm{\θ}\right)}^2\+r^2{\sin \left(\mathrm{\θ}\right)}^2\right)}^{3\/2}}$

$\stackrel{\text{assuming positive}}{\to }$

$\frac{{\mathrm{D}}_{1\,1}\left(F\right)\left(r\,\arctan \left(\sin \left(\mathrm{\θ}\right)\,\cos \left(\mathrm{\θ}\right)\right)\right)r^2\+{\mathrm{D}}_1\left(F\right)\left(r\,\arctan \left(\sin \left(\mathrm{\θ}\right)\,\cos \left(\mathrm{\θ}\right)\right)\right)r\+{\mathrm{D}}_{2\,2}\left(F\right)\left(r\,\arctan \left(\sin \left(\mathrm{\θ}\right)\,\cos \left(\mathrm{\θ}\right)\right)\right)}{r^2}$

$\stackrel{\text{assuming real range}}{\to }$

$\frac{{\mathrm{D}}_{1\,1}\left(F\right)\left(r\,\mathrm{\θ}\right)r^2\+{\mathrm{D}}_1\left(F\right)\left(r\,\mathrm{\θ}\right)r\+{\mathrm{D}}_{2\,2}\left(F\right)\left(r\,\mathrm{\θ}\right)}{r^2}$

$\stackrel{\text{to diff}}{\to }$

$\frac{\left(\frac{{\partial }^2}{\partial r^2}F\left(r\,\mathrm{\θ}\right)\right)r^2\+\left(\frac{\partial }{\partial r}F\left(r\,\mathrm{\θ}\right)\right)r\+\frac{{\partial }^2}{\partial {\mathrm{\θ}}^2}F\left(r\,\mathrm{\θ}\right)}{r^2}$

$\stackrel{\text{expand}}{\=}$

$\frac{{\partial }^2}{\partial r^2}F\left(r\,\mathrm{\θ}\right)\+\frac{\frac{\partial }{\partial r}F\left(r\,\mathrm{\θ}\right)}{r}\+\frac{\frac{{\partial }^2}{\partial {\mathrm{\θ}}^2}F\left(r\,\mathrm{\θ}\right)}{r^2}$

Define $f\left(x\,y\right)\=F\left(r\,\mathrm{\θ}\right)\=F\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)$

$f≔F\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)\:$

Use diff to calculate $f_{\mathrm{xx}}\+f_{\mathrm{yy}}$ and apply simplify

$q_1≔\mathrm{simplify}\left(\mathrm{diff}\left(f\,x\,x\right)\+\mathrm{diff}\left(f\,y\,y\right)\right)$

$\frac{{\mathrm{D}}_{1\,1}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)x^2\sqrt{x^2\+y^2}\+{\mathrm{D}}_{1\,1}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}{y}^2\+{\mathrm{D}}_1\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)x^2\+{\mathrm{D}}_1\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)y^2\+{\mathrm{D}}_{2\,2}\left(F\right)\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}}{{\left(x^2\+y^2\right)}^{3\/2}}$

Use eval to replace Cartesian with polar coordinates and apply simplify

$q_2≔\mathrm{simplify}\left(\mathrm{eval}\left(q_1\,\left[x\=r \cos \left(\mathrm{\θ}\right)\,y\=r \sin \left(\mathrm{\θ}\right)\right]\right)\right) assuming r\>0\,\mathrm{\θ}\colon\colon \mathrm{RealRange}\left(\mathrm{Open}\left(-\mathrm{\π}\right)\,\mathrm{\π}\right)$

$\frac{{\mathrm{D}}_{1\,1}\left(F\right)\left(r\,\mathrm{\θ}\right)r^2\+{\mathrm{D}}_1\left(F\right)\left(r\,\mathrm{\θ}\right)r\+{\mathrm{D}}_{2\,2}\left(F\right)\left(r\,\mathrm{\θ}\right)}{r^2}$

Use the convert command to change the differential-operator notation to partial-derivative notation, and use expand to split into separate terms

$\mathrm{expand}\left(\mathrm{convert}\left(q_2\,\mathrm{diff}\right)\right)$

$\frac{{\partial }^2}{\partial r^2}F\left(r\,\mathrm{\θ}\right)\+\frac{\frac{\partial }{\partial r}F\left(r\,\mathrm{\θ}\right)}{r}\+\frac{\frac{{\partial }^2}{\partial {\mathrm{\θ}}^2}F\left(r\,\mathrm{\θ}\right)}{r^2}$