Problem 24 - Essential Calculus Early Transcendentals

Question

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(y^{2}=x^{2}+4 z^{2}+4\)

Step-by-Step Solution

Answer

The surface is a hyperboloid of one sheet: \( \frac{y^2}{4} - \frac{x^2}{4} - z^2 = 1 \).

Step 1: Arrange Equation in Standard Form

The given equation is \(y^{2} = x^{2} + 4z^{2} + 4\). Our goal is to rewrite this equation to match one of the standard forms of conic sections. Start by rearranging the terms: \[ y^{2} - x^{2} - 4z^{2} = 4 \]

Step 2: Identify the Standard Form

The rearranged equation \( y^{2} - x^{2} - 4z^{2} = 4 \) resembles the standard form of a hyperboloid of one sheet which is \( \frac{y^2}{c^2} - \frac{x^2}{a^2} - \frac{z^2}{b^2} = 1 \). To match, divide the entire equation by 4:\[ \frac{y^{2}}{4} - \frac{x^{2}}{4} - z^{2} = 1 \]

Step 3: Classify the Surface

The equation \( \frac{y^2}{4} - \frac{x^2}{4} - z^2 = 1 \) is now in the standard form for a hyperboloid of one sheet, where \( a^2 = 4 \), \( b^2 = 4 \), and \( c^2 = 1 \).

Step 4: Sketch the Hyperboloid

To sketch the hyperboloid, note the following characteristics: 1. It is centered at the origin (0, 0, 0). 2. The z-axis is the axis of symmetry. 3. It opens along the y-axis. 4. The vertices are at (0, ±2, 0) and saddle points extend as hyperbolic sheets along the x and z axes.

Key Concepts

Hyperboloid

A hyperboloid is a fascinating surface found frequently in mathematics, especially within the realm of multivariable calculus. It comes in two types: hyperboloid of one sheet and hyperboloid of two sheets. The equation we are dealing with is a hyperboloid of one sheet, which has a unique appearance that can be visualized like a cooling tower of a power plant.

This type of hyperboloid is characterized by its ability to open along one axis and have hyperbolic curves in cross-sections parallel to the other axes. In the equation from our exercise, we find that the hyperboloid centers at the origin and opens along the y-axis, following the standard form:

• The general equation is \(\frac{y^2}{c^2} - \frac{x^2}{a^2} - \frac{z^2}{b^2} = 1\)
• Here, \(c^2=1\), \(a^2=4\), and \(b^2=4\)
• The hyperboloid is symmetric with respect to the xy-plane and the yz-plane
• It extends infinitely along all three axes, primarily expanding along the y-axis

By understanding these properties, students can visualize and sketch this three-dimensional surface more effectively. Recognizing the hyperboloid in this context clarifies how the structure can appear in practical applications.

Conic Sections

Conic sections describe the curves obtained by intersecting a plane with a cone. These include circles, ellipses, parabolas, and hyperbolas. In three dimensions, similar structures emerge, providing insight into the surfaces such as hyperboloids.

In multivariable calculus, conic sections help in recognizing patterns in these surfaces. When looking at our equation, the transformation into a hyperboloid of one sheet is possible because of the inherent nature of conic sections:

• Conic sections are defined by second-degree polynomial equations
• By rearranging and classifying these equations, we can identify the type of surface (e.g., hyperboloid of one sheet)
• The hyperboloid obtained is essentially a three-dimensional extension of hyperbolic curves

The step of substituting values and rearranging helps categorize these equations. Understanding this categorization allows students to identify surfaces by sight, sort them by their characteristics, and apply crucial concepts from 2D conic sections to 3D coordinate systems.

Coordinate Geometry

Coordinate geometry bridges algebra and geometry, enabling us to translate equations into compelling visual representations. In multivariable calculus, it's especially useful for visualizing and analyzing surfaces.

Our given hyperboloid example illustrates how a simple algebraic equation can describe complex geometry:

• The initial step involved rewriting the equation in standard form, essential for recognizing it as a hyperboloid
• By dividing coefficients, the equation was transformed into a cleaner form for a straightforward geometric interpretation
• The resulting surface, a hyperboloid, is described not just by its formula but by its symmetry and spatial orientation

Coordinate geometry helps students put together a coordinate grid to graph these surfaces, understand their properties, and solve real-world problems. It turns abstract equations into tangible forms students can observe and reason with, thereby making abstract mathematical concepts graspable through visual aids.