Problem 87 - Calculus Early Transcendentals

Question

Identify and briefly describe the surfaces defined by the following equations. $$y^{2}-z^{2}=2$$

Step-by-Step Solution

Answer

Answer: The surface represented by the equation $$y^2 - z^2 = 2$$ is a hyperboloid of two sheets with no rotation along the x-axis and with equal semi-axes along the y and z directions.

Step 1: Analyze the equation

We are given the equation $$y^{2}-z^{2}=2$$ and by comparing with the standard form of a hyperbolic equation, we observe that it represents a hyperboloid. In general, a hyperboloid equation can be written as: $$\frac{x^{2}}{a^{2}}\pm \frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1$$ However, in our equation there is no x-component, so this is a hyperboloid of two sheets since the sum of the square terms is equal to the constant, 2.

Step 2: Identify the surface

Since we found that the equation represents a hyperboloid of two sheets, we can now write the equation in its standard form. From the given equation $$y^{2}-z^{2}=2$$, we can write it as: $$\frac{z^{2}}{2}-\frac{y^{2}}{2}=1$$ So, in this standard form, we can identify that \(a^2 = b^2=2\) and there is no x term which implies no rotation along the x-axis.

Step 3: Briefly describe the surface

The surface represented by the equation $$y^{2}-z^{2}=2$$ is a hyperboloid of two sheets with no rotation along the x-axis and with equal semi-axes along the y and z directions. The surface consists of two separate curved surfaces that are infinite and open along the y and z directions, symmetric with respect to the yz-plane.