Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

The hyperbola \(2 x^{2}-z^{2}=2\) in the \(x z\)-plane is revolved about the \(z\)-axis. Write the equation of the resulting surface in cylindrical coordinates.

Step-by-Step Solution

Answer

The equation in cylindrical coordinates is \( r^2 = \frac{z^2}{2} + 1 \).

Step 1: Convert Cartesian Equation to Cylindrical Form

The original hyperbola equation in the Cartesian coordinates is \(2x^2 - z^2 = 2\). In cylindrical coordinates, \(x = r \cos\theta\), so we substitute \(x\) with \(r \cos\theta\) and rewrite the equation as \(2(r \cos\theta)^2 - z^2 = 2\).

Step 2: Simplify the Equation

Simplify \(2(r \cos \theta)^2 - z^2 = 2\) to get \(2r^2 \cos^2\theta - z^2 = 2\). Note that only the radial variable \(r\) and \(z\) should appear in the equation because the surface model should be independent of \(\theta\).

Step 3: Account for the Surface Revolution

Since the curve is revolved around the \(z\)-axis, the relationship involves \(r\) without the \(\theta\) dependence. Convert \(r^2\cos^2\theta\) to \(r^2\) in the equation to represent the radial symmetry in cylindrical coordinates: \(r^2 = \frac{z^2 + 2}{2}\).

Step 4: Final Equation of the Surface

The final equation in cylindrical coordinates is \( r^2 = \frac{z^2}{2} + 1 \). This represents the surface obtained when the hyperbola is revolved about the \(z\)-axis.

Key Concepts

Hyperbola

A hyperbola is a type of conic section that appears as two open curves, resembling two mirrored arches. The standard form of a hyperbola equation in Cartesian coordinates is given by:

• Horizontal Hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• Vertical Hyperbola: \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \)

The hyperbola involved in this problem is expressed as \(2x^2 - z^2 = 2\), which is aligned with the \(xz\)-plane. A hyperbola is characterized by its two branches that are symmetric with respect to its center, and they open outwards indefinitely. Key geometric elements related to a hyperbola include its vertices, foci, asymptotes, and transverse and conjugate axes. In the context of our problem, the hyperbola is not expressed by \(y\)-coordinates, and the revolution occurs around the \(z\)-axis, making it uniquely a three-dimensional concern.

Surface Revolution

Surface revolution occurs when a two-dimensional curve is rotated around an axis, resulting in a three-dimensional shape or surface. For example, if you revolve a circle around one of its diameters, you obtain a sphere. In our exercise, the hyperbola in the \(xz\)-plane is revolved around the \(z\)-axis. This operation translates the curve into a surface that is symmetric along the axis of revolution.

To visualize this, imagine taking a paper curve and twisting it around a stick. The resulting surface will be smooth and continuous, expanding outwards along the axis. What makes this important from a mathematical perspective is that the resulting 3D surface can often be described by transformations into other coordinate systems, such as cylindrical coordinates, to exploit their symmetrical nature.

Parametric Equations

Parametric equations are sets of equations that express the coordinates of the points making up a geometric figure, using one or more independent parameters. Essentially, instead of describing an object with \(x\) and \(y\) coordinates directly, these equations use a third variable, often \(t\), \(\theta\), or another parameter, to provide a detailed description.

In the context of our problem, although it is not directly solved using parametric form, the idea plays a subtle role in the cylindrical coordinates. If needed, the surface can be expressed parametrically by accounting for \(r\), \(\theta\), and \(z\):

• \( r = r(\theta, z) \)
• \( \theta = \theta(t) \)
• \( z = z(t) \)

This implies that every point on the surface can be traced using these parameters, giving a full picture of the surface with respect to a chosen parameter. Parametric equations are especially helpful in computer graphics and fields requiring dynamic modeling.

Coordinate Transformation

Coordinate transformation is the process of converting the coordinates of a point or geometric figure from one coordinate system to another. This can simplify mathematical problems by exploiting the symmetries and properties of different systems.

In our exercise, the transformation from Cartesian to cylindrical coordinates makes it easier to handle the symmetry of revolving surfaces around the \(z\)-axis. In cylindrical coordinates:

• \( x = r \cos\theta \)
• \( y = r \sin\theta \)
• \( z = z \)

Such transformations allow dealing with problems more efficiently by adapting to a system that aligns neatly with the object's geometry. The change was pivotal in simplifying the hyperbola's equation to effectively describe the 3D surface produced by the revolution. Utilizing cylindrical coordinates made the minor dependency on \(\theta\) clear and simplified the description of the surface's radial properties.