Question

Describe the level surfaces of:

f(x, y, z) = x² – y²– z²

Short answer

Answer is not given in he drive.

Step-by-step solution

Defining Level Surfaces:

\(\begin{array}{l}{\bf{f}}\left( {{\bf{x}},{\rm{ }}{\bf{y}},{\rm{ }}{\bf{z}}} \right) &= k\\{{\bf{x}}^{\bf{2}}} - {\rm{ }}{{\bf{y}}^{\bf{2}}} - {z^2} &= k\end{array}\)

For k > 0:

\(\begin{array}{l}{{\bf{x}}^{\bf{2}}} - {\rm{ }}{{\bf{y}}^{\bf{2}}} - {z^2} &= k\\\frac{{{x^2}}}{k} - \frac{{{y^2}}}{k} - \frac{{{z^2}}}{k} &= 1\\\frac{{{x^2}}}{{{k^2}}} - \frac{{{y^2}}}{{{k^2}}} - \frac{{{z^2}}}{{{k^2}}} &= 1\end{array}\)

This represents the following graph:

For k < 0

\(\begin{array}{l} \Rightarrow {{\bf{x}}^{\bf{2}}} - {\rm{ }}{{\bf{y}}^{\bf{2}}} - {z^2} &= - k\\ - \frac{{{x^2}}}{{{k^2}}} + \frac{{{y^2}}}{{{k^2}}} + \frac{{{z^2}}}{{{k^2}}} &= 1\end{array}\)

This represents the hyperboloid as shown: