Problem 42 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-44.\) ASSORTED $$y^{2}-x^{2}-z^{2}=1$$

Step-by-Step Solution

Answer

The surface is a hyperboloid of one sheet centered on the y-axis.

Step 1: Identify the Standard Form

The given equation is \( y^2 - x^2 - z^2 = 1 \). This matches the standard form \( y^2 - x^2 - z^2 = k \), which describes a hyperboloid of one sheet if \( k > 0 \). In our case, \( k = 1 \), so it is a hyperboloid of one sheet.

Step 2: Determine the Axes of Symmetry

The equation \( y^2 - x^2 - z^2 = 1 \) shows symmetry around the y-axis, as the y-term has the positive sign while x and z terms are subtracted. This means the hyperboloid is centered on and opens around the y-axis.

Step 3: Analyze the Trace in Coordinate Planes

To understand the 3D structure, consider its intersections with standard planes. For example, setting \( x = 0 \) results in \( y^2 - z^2 = 1 \), a hyperbola in the yz-plane. Setting \( z = 0 \) results in \( y^2 - x^2 = 1 \), a hyperbola in the yx-plane. Setting \( y = 0 \), gives \( -x^2 - z^2 = 1 \), which is not possible, indicating no real traces in the xz-plane.

Step 4: Sketch the Surface

Start with the yz-plane: draw a hyperbola with the y-axis as the transverse axis (centered at the origin with vertices moving along the y-axis). In the yx-plane, also draw a hyperbola with the y-axis as the transverse axis. Recognize that vertically, along the y-axis, the structure expands outwards as y increases or decreases, forming a complex 3D structure resembling two connected hyperbolic sheets around the y-axis.

Key Concepts

Hyperboloid of One Sheet

A hyperboloid of one sheet is a fascinating type of three-dimensional surface. It's one of many quadric surfaces, which are the three-dimensional cousins of conic sections like circles and parabolas. The general equation for a hyperboloid of one sheet can be expressed in the standard form:

• For a hyperboloid aligned along the z-axis: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \)
• For a hyperboloid aligned along the y-axis: \( \frac{y^2}{a^2} - \frac{x^2}{b^2} - \frac{z^2}{c^2} = 1 \)

The surface is named for its unique shape, which resembles two connected hyperbolic sheets. These sheets open around the axis with the positive term (like the y-axis in our exercise). This characteristic makes the hyperboloid of one sheet distinct and easy to recognize. It visually appears as two semi-elliptical blocks connected in a way that curves smoothly, forming a continuous surface.

Three-Dimensional Surfaces

Three-dimensional surfaces in mathematics can be complex but are essential for understanding multivariable calculus and several applied sciences. A surface in three dimensions is defined by an equation in three variables, typically x, y, and z. These surfaces can include planes, cylinders, spheres, ellipsoids, and hyperboloids, among others.
These surfaces have different shapes and properties based on their equations:

• Planes are the simplest, described by linear equations like \( ax + by + cz = d \).
• Spheres have perfectly round surfaces, governed by equations such as \( x^2 + y^2 + z^2 = r^2 \).
• Ellipsoids are extended spheres and have equations like \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\).

For an equation like the hyperboloid of one sheet in our exercise, understanding these surfaces place a foundational role in visualizing how they behave, particularly in understanding spatial symmetry and trace patterns.

Coordinate Plane Traces

Coordinate plane traces offer insights into how a three-dimensional surface like a hyperboloid interacts with two-dimensional planes. By setting one of the coordinates to zero, you can analyze the resulting two-dimensional curve, a method that provides a cross-sectional view of the surface.
For the hyperboloid of one sheet equation \( y^2 - x^2 - z^2 = 1 \):

• Setting \( x = 0 \) gives the trace \( y^2 - z^2 = 1 \), a hyperbola in the yz-plane.
• Setting \( z = 0 \) gives \( y^2 - x^2 = 1 \), also a hyperbola, but this time in the yx-plane.
• Setting \( y = 0 \) results in \( -x^2 - z^2 = 1 \), which has no real solutions and hence, no real trace in the xz-plane.

Analyzing these traces helps you build a mental picture of how the surface exists in space, offering clues on the orientation and shape of the entire structure of the hyperboloid of one sheet.