Problem 37 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-76\) $$ \left(y^{2} / 4\right)+\left(z^{2} / 9\right)-\left(x^{2} / 4\right)=1 $$

Step-by-Step Solution

Answer

The surface is a hyperboloid of two sheets oriented along the x-axis.

Step 1: Identify the Surface Type

The given equation \( \frac{y^2}{4} + \frac{z^2}{9} - \frac{x^2}{4} = 1 \) can be identified as the equation of a hyperboloid of two sheets. This is because the sum of a positive quadratic term and a negative quadratic term equals a constant.

Step 2: Rewrite for Clarity

Re-write the given equation in a more recognizable form: \[ \frac{y^2}{4} + \frac{z^2}{9} - \frac{x^2}{4} = 1. \] This more explicitly shows the positive and negative contributions from the terms involving \(y\), \(z\), and \(x\).

Step 3: Determine Orientation

Notice how the \(x\)-term, \(-\frac{x^2}{4}\), is negative in the equation. This indicates that the axis of symmetry for the hyperboloid is along the \(x\)-axis.

Step 4: Identify Extent of Curvature

The terms \(\frac{y^2}{4}\) and \(\frac{z^2}{9}\) determine the stretching of the hyperboloid along the respective \(y\) and \(z\) directions. The coefficients \(4\) and \(9\) are inversely related to the sizes of the half-widths along these axes.

Step 5: Sketching the Hyperboloid

Begin by sketching the intersection with the plane \(x = 0\), which gives \(\frac{y^2}{4} + \frac{z^2}{9} = 1\). This is the equation of an ellipse in the \(yz\)-plane. As \(x\) increases positively or negatively, this ellipse moves outward, creating the classic two-sheet hyperboloid shape along the \(x\)-axis.

Key Concepts

Hyperboloid

The hyperboloid is a fascinating type of surface that can appear in different forms, either as a one-sheet or a two-sheet hyperboloid. In the equation \( \frac{y^2}{4} + \frac{z^2}{9} - \frac{x^2}{4} = 1 \), we encounter a hyperboloid of two sheets.
This specific type of hyperboloid is characterized by:

• Two separate regions (sheets) extending along the axis of symmetry.
• Each sheet resembles a surface that curves outward and looks like a dish.
• The equation structure, specifically having one negative squared term, dictates the type of hyperboloid and its orientation.

Knowing that the hyperboloid is of two sheets helps in understanding its spatial configuration and how it fits into the world of quadric surfaces.

Quadric Surfaces

Quadric surfaces are a broad category of surfaces defined by quadratic equations in three variables. They include various familiar shapes such as ellipsoids, hyperboloids, paraboloids, among others.
In the provided equation \( \frac{y^2}{4} + \frac{z^2}{9} - \frac{x^2}{4} = 1 \), the surface is a specific type of quadric surface known as a hyperboloid. The equation features:

• A combination of three squared terms, each divided by a constant.
• The presence of both positive and negative signs among these terms—vital in shaping the surface's form.
• Differences in coefficients reflecting how the surface stretches in different directions.

Understanding quadric surfaces means recognizing these patterns and signs, which guide us in predicting the surface's shape and orientation without necessarily needing to sketch it immediately.

Surface Sketching

Sketching surfaces like the hyperboloid involves determining how the surface interacts with specific planes. This helps visualize the three-dimensional shape effectively. Here's a simplified approach:

• Identify key planes to intersect the surface, such as \(x = 0\), which in this case gives us an ellipse \( \frac{y^2}{4} + \frac{z^2}{9} = 1 \) in the \(yz\)-plane.
• Examine cross-sections perpendicular to different axes to understand the overall shape.
• As the hyperboloid extends along the \(x\)-axis, observing these cross-sections shows the forms they take as they move away from the \(yz\)-plane.

By following these steps, once complex three-dimensional surfaces become easier to comprehend and draw, making it possible to predict shapes and structures that otherwise remain abstract.

Equation Identification

Identifying the equation of a surface is like solving a puzzle. It involves analyzing terms, signs, and coefficients to classify the surface correctly. To identify a hyperboloid among quadric surfaces:

• Look for a mix of positive and negative terms. Here, the \(-\frac{x^2}{4}\) is negative, indicating the axis of symmetry.
• Notice the constant on the right side of the equation. If it's non-zero, be meticulous in considering the types of surfaces possible.
• Verify coefficients of other terms to determine if any changes in signs or values skew the surface differently from the typical models.

Mastering equation identification supports deeper understanding in multivariable calculus, as it allows visualization and classification of different surfaces, aiding further problem-solving.