3639.sagews

s(x,y,z,r)=(x*r*r/(x*x+y*y+z*z),y*r*r/(x*x+y*y+z*z),z*r*r/(x*x+y*y+z*z))

jacobian(s, [x,y,z])(x,y,z)

\newcommand{\Bold}[1]{\mathbf{#1}}\left($$\begin{array}{rrr} -\frac{2 \, r^{2} x^{2}}{{\left(x^{2} + y^{2} + z^{2}\right)}^{2}} + \frac{r^{2}}{x^{2} + y^{2} + z^{2}} & -\frac{2 \, r^{2} x y}{{\left(x^{2} + y^{2} + z^{2}\right)}^{2}} & -\frac{2 \, r^{2} x z}{{\left(x^{2} + y^{2} + z^{2}\right)}^{2}} \\ -\frac{2 \, r^{2} x y}{{\left(x^{2} + y^{2} + z^{2}\right)}^{2}} & -\frac{2 \, r^{2} y^{2}}{{\left(x^{2} + y^{2} + z^{2}\right)}^{2}} + \frac{r^{2}}{x^{2} + y^{2} + z^{2}} & -\frac{2 \, r^{2} y z}{{\left(x^{2} + y^{2} + z^{2}\right)}^{2}} \\ -\frac{2 \, r^{2} x z}{{\left(x^{2} + y^{2} + z^{2}\right)}^{2}} & -\frac{2 \, r^{2} y z}{{\left(x^{2} + y^{2} + z^{2}\right)}^{2}} & -\frac{2 \, r^{2} z^{2}}{{\left(x^{2} + y^{2} + z^{2}\right)}^{2}} + \frac{r^{2}}{x^{2} + y^{2} + z^{2}} \end{array}$$\right)

J = jacobian(s, [x,y,z])(x,y,z)

O = (J/(r^2/(x*x+y*y+z*z))).simplify_rational()

(O.transpose()*O).simplify_rational()

\newcommand{\Bold}[1]{\mathbf{#1}}\left($$\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}$$\right)