$f_x\=F_r r_x\+F_t t_x$

and

$f_y\=F_r r_y\+F_t t_y$

$f_{\mathrm{xx}}\=\left(F_{\mathrm{rr}} r_x\+F_{\mathrm{rt}} t_x\right)r_x\+F_r r_{\mathrm{xx}}\+\left(F_{\mathrm{tr}} r_x\+F_{\mathrm{tt}} t_x\right)t_x\+F_t t_{\mathrm{xx}}$ 

$f_{\mathrm{yy}}\=\left(F_{\mathrm{rr}} r_y\+F_{\mathrm{rt}} t_y\right)r_y\+F_r r_{\mathrm{yy}}\+\left(F_{\mathrm{tr}} r_y\+F_{\mathrm{tt}} t_y\right)t_y\+F_t t_{\mathrm{yy}}$

$r_x\=\frac{x}{\sqrt{x^2\+y^2}}\=\frac{r \cos \left(t\right)}{r}\=\cos \left(t\right)$

$r_{\mathrm{xx}}\=\frac{y^2}{{\left(x^2+y^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\=\frac{r^2{\sin }^2\left(t\right)}{r^3}\=\frac{{\sin }^2\left(t\right)}{r}$

$r_y\=\frac{y}{\sqrt{x^2\+y^2}}\=\frac{r \sin \left(t\right)}{r}\=\sin \left(t\right)$

$r_{\mathrm{yy}}\=\frac{x^2}{{\left(x^2+y^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\=\frac{r^2{\cos }^2\left(t\right)}{r^3}\=\frac{{\cos }^2\left(t\right)}{r}$

$t_x\=\frac{-y\/x^2}{1\+{\left(y\/x\right)}^2}\=-\frac{y}{x^2\+y^2}\=-\frac{\sin \left(t\right)}{r}$

$t_{\mathrm{xx}}\=\frac{2yx}{{\left(x^2+y^2\right)}^2}\=\frac{2 r^2\sin \left(t\right)\cos \left(t\right)}{r^4}\=\frac{2 \sin \left(t\right)\cos \left(t\right)}{r^2}$

$t_y\=\frac{1\/x}{1\+{\left(y\/x\right)}^2}\=\frac{x}{x^2\+y^2}\=\frac{\cos \left(t\right)}{r}$

$t_{\mathrm{yy}}\=-\frac{2yx}{{\left(x^2+y^2\right)}^2}\=-\frac{2 \sin \left(t\right)\cos \left(t\right)}{r^2}$

Table 4.3.30(a)   First and second derivatives of $r\left(x\,y\right)$ and $t\left(x\,y\right)$ 

$r_x^2\+r_y^2\={\cos }^2\left(t\right)\+{\sin }^2\left(t\right)\=1$

$t_x r_x\+t_y r_y\=\left(-\frac{\sin \left(t\right)}{r}\right)\cos \left(t\right)\+\left(\frac{\cos \left(t\right)}{r}\right)\sin \left(t\right)\=\frac{0}{r}\=0$

$t_x^2\+t_y^2\={\left(-\frac{\sin \left(t\right)}{r}\right)}^2\+{\left(\frac{\cos \left(t\right)}{r}\right)}^2\=\frac{1}{r^2}$

$r_{\mathrm{xx}}\+r_{\mathrm{yy}}\=\frac{{\sin }^2\left(t\right)}{r}\+\frac{{\cos }^2\left(t\right)}{r}\=\frac{1}{r}$

$t_{\mathrm{xx}}\+t_{\mathrm{yy}}\=\frac{2 \sin \left(t\right)\cos \left(t\right)}{r^2}\+\left(-\frac{2 \sin \left(t\right)\cos \left(t\right)}{r^2}\right)\=0$

Table 4.3.30(b)   Coefficients of $f_{\mathrm{xx}}\+f_{\mathrm{yy}}$

Assign the starting composition statement the name $a$ 

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

Assign the sum of second partial derivatives the name $b$ 

$b\=\frac{{\partial }^2}{{\partial }^{}{x}^2} a\+\frac{{\partial }^2}{{\partial }^{}{y}^2} a$$\stackrel{\text{assign}}{\to }$

Replace $x$ and $y$ with $r \cos \left(t\right)$ and $r \sin \left(t\right)$, respectively, and simplifying

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

${\mathrm{D}}_{1\,1}\left(f\right)\left(r\cos \left(t\right)\,r\sin \left(t\right)\right)\+{\mathrm{D}}_{2\,2}\left(f\right)\left(r\cos \left(t\right)\,r\sin \left(t\right)\right)\=\frac{\left(\frac{{\mathrm{D}}_{1\,1}\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right)r\cos \left(t\right)}{\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}}-\frac{{\mathrm{D}}_{1\,2}\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right)\sin \left(t\right)}{r{\cos \left(t\right)}^2\left(1\+\frac{{\sin \left(t\right)}^2}{{\cos \left(t\right)}^2}\right)}\right)r\cos \left(t\right)}{\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}}-\frac{{\mathrm{D}}_1\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right)r^2{\cos \left(t\right)}^2}{{\left(r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2\right)}^{3\/2}}\+\frac{2{\mathrm{D}}_1\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right)}{\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}}-\frac{\left(\frac{{\mathrm{D}}_{1\,2}\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right)r\cos \left(t\right)}{\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}}-\frac{{\mathrm{D}}_{2\,2}\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right)\sin \left(t\right)}{r{\cos \left(t\right)}^2\left(1\+\frac{{\sin \left(t\right)}^2}{{\cos \left(t\right)}^2}\right)}\right)\sin \left(t\right)}{r{\cos \left(t\right)}^2\left(1\+\frac{{\sin \left(t\right)}^2}{{\cos \left(t\right)}^2}\right)}\+\frac{2{\mathrm{D}}_2\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right)\sin \left(t\right)}{r^2{\cos \left(t\right)}^3\left(1\+\frac{{\sin \left(t\right)}^2}{{\cos \left(t\right)}^2}\right)}-\frac{2{\mathrm{D}}_2\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right){\sin \left(t\right)}^3}{r^2{\cos \left(t\right)}^5{\left(1\+\frac{{\sin \left(t\right)}^2}{{\cos \left(t\right)}^2}\right)}^2}\+\frac{\left(\frac{{\mathrm{D}}_{1\,1}\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right)r\sin \left(t\right)}{\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}}\+\frac{{\mathrm{D}}_{1\,2}\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right)}{r\cos \left(t\right)\left(1\+\frac{{\sin \left(t\right)}^2}{{\cos \left(t\right)}^2}\right)}\right)r\sin \left(t\right)}{\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}}-\frac{{\mathrm{D}}_1\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right)r^2{\sin \left(t\right)}^2}{{\left(r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2\right)}^{3\/2}}\+\frac{\frac{{\mathrm{D}}_{1\,2}\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right)r\sin \left(t\right)}{\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}}\+\frac{{\mathrm{D}}_{2\,2}\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right)}{r\cos \left(t\right)\left(1\+\frac{{\sin \left(t\right)}^2}{{\cos \left(t\right)}^2}\right)}}{r\cos \left(t\right)\left(1\+\frac{{\sin \left(t\right)}^2}{{\cos \left(t\right)}^2}\right)}-\frac{2{\mathrm{D}}_2\left(F\right)\left(\sqrt{r^2{\cos \left(t\right)}^2\+r^2{\sin \left(t\right)}^2}\,\arctan \left(r\sin \left(t\right)\,r\cos \left(t\right)\right)\right)\sin \left(t\right)}{r^2{\cos \left(t\right)}^3{\left(1\+\frac{{\sin \left(t\right)}^2}{{\cos \left(t\right)}^2}\right)}^2}$

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

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

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

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

$\stackrel{\text{right hand side}}{\to }$

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

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

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

Notational simplifications

$$\begin{gathered} \mathrm{interface}\left(\mathrm{typesetting}\=\mathrm{extended}\right)\: \\ \mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[r\left(x\,y\right)\,t\left(x\,y\right)\right]\right)\: \\ \mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}\=\mathrm{true}\right)\: \end{gathered}$$

Form the sum $f_{\mathrm{xx}}\+f_{\mathrm{yy}}$ 

$q_1≔\mathrm{collect}\left(\mathrm{diff}\left(F\left(r\,t\right)\,x\,x\right)\+\mathrm{diff}\left(F\left(r\,t\right)\,y\,y\right)\,\mathrm{D}\right)$

$q_1≔\left(r_{x,x}+r_{y,y}\right){\mathrm{D}}_1\left(F\right)\left(r\,t\right)+\left(t_{x,x}+t_{y,y}\right){\mathrm{D}}_2\left(F\right)\left(r\,t\right)+\left({\left(r_x\right)}^2+{\left(r_y\right)}^2\right){\mathrm{D}}_{1,1}\left(F\right)\left(r\,t\right)+\left(2\left(r_x\right)\left(t_x\right)+2\left(r_y\right)\left(t_y\right)\right){\mathrm{D}}_{1,2}\left(F\right)\left(r\,t\right)+\left({\left(t_x\right)}^2+{\left(t_y\right)}^2\right){\mathrm{D}}_{2,2}\left(F\right)\left(r\,t\right)$

Define $r\=r\left(x\,y\right)\=\sqrt{x^2\+y^2}$ and $t\=t\left(x\,y\right)\=\arctan \left(y\/x\right)$ 

Evaluate the coefficients in the expression for $f_{\mathrm{xx}}\+f_{\mathrm{yy}}$ 

$\mathrm{simplify}\left(\mathrm{diff}\left(R\,x\,x\right)\+\mathrm{diff}\left(R\,y\,y\right)\right)$ = $\frac{1}{\sqrt{x^2+y^2}}$

$\mathrm{simplify}\left(\mathrm{diff}\left(T\,x\,x\right)\+\mathrm{diff}\left(T\,y\,y\right)\right)$ = $0$

$\mathrm{simplify}\left(\mathrm{diff}{\left(R\,x\right)}^2\+\mathrm{diff}{\left(R\,y\right)}^2\right)$ = $1$

$\mathrm{simplify}\left(2 \mathrm{diff}\left(R\,x\right) \mathrm{diff}\left(T\,x\right)\+2 \mathrm{diff}\left(R\,y\right) \mathrm{diff}\left(T\,y\right)\right)$ = $0$

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

Rewrite the expression for $f_{\mathrm{xx}}\+f_{\mathrm{yy}}$ 

$f_{\mathrm{xx}}\+f_{\mathrm{yy}}\=\frac{1}{r}{F}_r\+F_{\mathrm{rr}}\+\frac{1}{r^2}{F}_{\mathrm{tt}}$ 

Obtain the Laplacian in polar coordinates

$\mathrm{Typesetting}:-\mathrm{Unsuppress}\left(\left[r\left(x\,y\right)\,t\left(x\,y\right)\right]\right)\:$

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(F\left(r\,t\right)\right)\:$

Apply the Laplacian command from the Student VectorCalculus package

$\mathrm{expand}\left(\mathrm{Student}:-\mathrm{VectorCalculus}:-\mathrm{Laplacian}\left(F\left(r\,t\right)\,\mathrm{coords}\=\mathrm{polar}\left[r\,t\right]\right)\right)$

$\frac{F_r}{r}+F_{r,r}+\frac{F_{t,t}}{r^2}$