Answer to Question #106743 in Analytic Geometry for Nikesh gautam pandit ji

Question

Express the following surfaces in spherical coordinates

i) xz = 3
ii) x^2+y^2-z^2=1

Answer 1

Formulas for converting Euclidean coordinates (x,y,z) to spherical coordinates $(r,\theta,\phi)$ have the form:

$x=r\cdot sin\theta \cdot cos\phi\\y=r\cdot sin\theta \cdot sin\phi\\z=r\cdot cos\theta$

The definition of spherical coordinates and their relation to Euclidean coordinates is shown in the figure

The surface (i) $xz=3$ in spherical coordinates will be $x\cdot z=r\cdot sin\theta \cdot cos\phi \cdot r\cdot cos\theta=r^2\cdot cos\theta \cdot sin\theta\cdot cos\phi=\frac{1}{2}r^2 sin(2\theta)cos\phi$

(i) $r^2 sin(2\theta)cos\phi=6$

The surface (ii) $x^2+y^2-z^2=1$ in spherical coordinates will be

$x^2+y^2-z^2=(r\cdot sin\theta \cdot cos\phi)^2+(r\cdot sin\theta \cdot sin\phi)^2-(r\cdot cos\theta )^2=\\r^2\cdot(sin^2\theta\cdot(cos^2\phi+sin^2\phi)-cos^2\theta)=r^2\cdot(sin^2\theta- cos^2\theta)=-r^2\cdot cos(2\theta)$

(ii) $r^2\cdot cos(2\theta)=-1$

In deriving these formulas, we used relations for the trigonometric functions of the double angle. The last surface (ii) does not depend on the angle $\phi$ and thus is the surface of rotation (revolution) around the Z axis.

Answer: The surfaces is expressed in spherical coordinates as

(i) $r^2 \cdot sin(2\theta)\cdot cos\phi=6$

(ii) $r^2\cdot cos(2\theta)=-1$