Spherical Coordinates — Question 4

Question 4 – Solution

Strategy Relate the polar angle to cylindrical radius and height: $r=\rho\sin\phi$, $z=\rho\cos\phi$.

Conversion At $\phi=\pi/4$, $$\frac{r}{z}=\tan\phi=1,$$ so $\boxed{z=r}$ in cylindrical coordinates. Therefore $$\boxed{z=\sqrt{x^2+y^2}}.$$

Geometry Because $\phi=\pi/4<\pi/2$, $z=\rho\cos\phi\ge 0$. Thus this is the upper nappe only. Squaring to $z^2=x^2+y^2$ without retaining $z\ge 0$ would incorrectly add the lower nappe.