Initialize

Loading Student:-VectorCalculus

Define the scalar field $f$ as a function

$f\left(x\,y\right)\=x\/y\+y\/x$$\stackrel{\text{assign as function}}{\to }$$f$

Change to polar coordinates

$f\left(r \cos \left(\mathrm{\θ}\right)\,r \sin \left(\mathrm{\θ}\right)\right)$ = $\frac{\cos \left(\mathrm{\theta }\right)}{\sin \left(\mathrm{\theta }\right)}+\frac{\sin \left(\mathrm{\theta }\right)}{\cos \left(\mathrm{\theta }\right)}$$\stackrel{\text{assign to a name}}{\to }$$F$$$

Obtain the Laplacian in polar coordinates then convert back to Cartesian coordinates

$F$

$\frac{\cos \left(\mathrm{\theta }\right)}{\sin \left(\mathrm{\theta }\right)}+\frac{\sin \left(\mathrm{\theta }\right)}{\cos \left(\mathrm{\theta }\right)}$

$\stackrel{\text{Laplacian}}{\to }$

$\frac{\frac{2\cos \left(\mathrm{\theta }\right)}{\sin \left(\mathrm{\theta }\right)}+\frac{2{\cos \left(\mathrm{\theta }\right)}^3}{{\sin \left(\mathrm{\theta }\right)}^3}+\frac{2\sin \left(\mathrm{\theta }\right)}{\cos \left(\mathrm{\theta }\right)}+\frac{2{\sin \left(\mathrm{\theta }\right)}^3}{{\cos \left(\mathrm{\theta }\right)}^3}}{r^2}$

$\stackrel{\text{evaluate at point}}{\to }$

$\frac{\frac{2x}{y}+\frac{2x^3}{y^3}+\frac{2y}{x}+\frac{2y^3}{x^3}}{x^2+y^2}$

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

$\frac{2x^4+2y^4}{y^3{x}^3}$

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

$\frac{2x}{y^3}+\frac{2y}{x^3}$

Initialize

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right)\:$

Define the scalar field $f$ as a function

$f≔\left(x\,y\right)\to x\/y\+y\/x\:$

Change to polar coordinates

$F≔f\left(r \cos \left(\mathrm{\θ}\right)\,r \sin \left(\mathrm{\θ}\right)\right)\:$

Apply the Laplacian command to obtain the Laplacian in polar coordinates

$\mathrm{LF}≔\mathrm{Laplacian}\left(F\,\mathrm{polar}\left[r\,\mathrm{\θ}\right]\right)$

$\frac{\frac{2\cos \left(\mathrm{\θ}\right)}{\sin \left(\mathrm{\θ}\right)}\+\frac{2{\cos \left(\mathrm{\θ}\right)}^3}{{\sin \left(\mathrm{\θ}\right)}^3}\+\frac{2\sin \left(\mathrm{\θ}\right)}{\cos \left(\mathrm{\θ}\right)}\+\frac{2{\sin \left(\mathrm{\θ}\right)}^3}{{\cos \left(\mathrm{\θ}\right)}^3}}{r^2}$

Restore Cartesian coordinates

$\mathrm{expand}\left(\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{LF}\,\left[r\=\sqrt{x^2\+y^2}\,\mathrm{\θ}\=\arctan \left(y\,x\right)\right]\right)\right)\right)$ = $\frac{2x}{y^3}\+\frac{2y}{x^3}$$$