$x^2 + y^2 - z^2 = 1$ Express the given surface in spherical coordinates

Question

I am really finding it difficult to understand and hence need help to find out what the answer is in . Please do it step wise as I am a noob in these type of questions.

Answer 1

The mappings from rectangular to spherical coordinates are

$$\begin{align} &x = \rho \sin(\phi) \cos(\theta) \\ &y = \rho \sin(\phi) \sin(\theta) \\ &z = \rho \cos(\phi) \end{align}$$

So we get:

$$\begin{align} &[\rho \sin(\phi) \cos(\theta)]^2 + [\rho \sin(\phi) \sin(\theta)]^2 - [\rho \cos(\phi)]^2 = 1 \\ &\Rightarrow \rho^2 \sin^2(\phi)\cos^2(\theta) + \rho^2 \sin^2(\phi)\sin^2(\theta) - \rho^2\cos^2(\phi) = 1 \\ &\Rightarrow \rho^2\sin^2(\phi)[\cos^2(\theta) + \sin^2(\theta)] - \rho^2 \cos^2(\phi) = 1 \\ &\Rightarrow \rho^2 \sin^2(\phi)[1] - \rho^2 \cos^2(\phi) = 1 \\ &\Rightarrow \rho^2[-\cos(2\phi)] = 1 \ \ \text{(By the trig. identity $\cos(2\phi) = \cos^2(\phi) - \sin^2(\phi)$.) } \\ &\Rightarrow \rho^2\cos(2\phi) = -1 \end{align}$$

Related: Find the equation in spherical coordinates of $x^2 + y^2 – z^2 = 4$.

For an excellent calculus textbook that covers this, see page 931 of Thomas' Calculus, 14th Edition, by Hass, Heil, and Weir.