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 \(r^2 = z^2 + 2\).

Step 1: Understand the Given Equation in Cartesian Coordinates

The given hyperbola is expressed in Cartesian coordinates as \(2x^2 - z^2 = 2\). This equation describes a hyperbola opening along the \(z\)-axis in the \(xz\)-plane. We need to transform this equation into cylindrical coordinates to find the equation of the surface generated when it is revolved around the \(z\)-axis.

Step 2: Transform Cartesian Coordinates to Cylindrical Coordinates

In cylindrical coordinates, the relation between Cartesian coordinates \((x, y, z)\) and cylindrical coordinates \((r, \theta, z)\) is given by \(x = r\cos\theta\), \(y = r\sin\theta\), and \(z = z\). Since \(x^2 + y^2 = r^2\), we can express \(2x^2\) in terms of \(r\) as \(2x^2 = 2(r^2 \cos^2\theta)\). Simplifying, it becomes \(2x^2 = 2r^2\cos^2\theta\). Substituting this into the hyperbola equation, we get \(2(r^2 \cos^2\theta) - z^2 = 2\).

Step 3: Rewrite and Simplify the Equation

Substitute the transformed expression into the original equation: \(2(r^2 \cos^2\theta) - z^2 = 2\). This simplifies to \(r^2 \cos^2\theta - \frac{z^2}{2} = 1\). We attempt to express this in a more recognizable form in cylindrical coordinates by factoring, yielding \(r^2(\cos^2\theta) = z^2 + 2\). Since the problem involves revolving around the \(z\)-axis, the \(\theta\)-dependence vanishes, leading to \(r = \sqrt{z^2 + 2}\).

Step 4: Write the Final Equation in Cylindrical Coordinates

The final equation in cylindrical coordinates becomes \(r^2 = z^2 + 2\) because, after revolving around the \(z\)-axis, every point is symmetric with respect to \(\theta\). This equation represents the surface generated by revolving the given hyperbola around the \(z\)-axis.

Key Concepts

Hyperbola

A hyperbola is a type of conic section formed by intersecting a double cone with a plane. In the equation provided, \(2x^2 - z^2 = 2\), we have a hyperbola situated in the \(xz\)-plane. This equation represents a hyperbolic curve because it features a difference of squares. Hyperbolas have two branches, and they open in directions dependent on the equation's terms. Here, the term \(-z^2\) results in the hyperbola opening along the \(z\)-axis.

Key features of a hyperbola include:

• Two branches that extend towards infinity
• An axis of symmetry, which in this case is the \(z\)-axis
• Defined by the property that the difference in distances to two fixed points (foci) is constant

Understanding the property and orientation of hyperbolas is important for recognizing how surfaces evolve when they undergo transformations.

Revolution of Surfaces

Revolution of surfaces is a process where a profile curve is revolved around an axis, creating a 3D surface. In this exercise, the curve described by the hyperbola \(2x^2 - z^2 = 2\) is revolved around the \(z\)-axis.

Key characteristics of surface revolution include:

• Generating symmetric shapes like cylinders, spheres, or paraboloids
• Each revolution results in a surface, typically ending in a smooth, continuous form
• The axis of revolution dictates the overall symmetry and contour of the resulting surface

When you revolve a profile curve such as our hyperbola, every point on the curve sweeps out a circle, leading to the form of the surface of revolution.

Coordinate Transformation

Coordinate transformation is crucial when dealing with different coordinate systems. It allows us to convert equations from one system to another, facilitating problem-solving. In this task, we transform the Cartesian coordinates \((x, y, z)\) to cylindrical coordinates \((r, \theta, z)\), essential for expressing the surface equation after the revolution.

The relationships are as follows:

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

Cylindrical coordinates prove beneficial when describing surfaces of revolution because they inherently embed the circular nature of such surfaces into the coordinate system, simplifying the math involved.

Surface Equation

The surface equation is the mathematical representation of a surface in space. For the revolved hyperbola, we arrive at the equation \(r^2 = z^2 + 2\) using cylindrical coordinates. This expression captures the geometry of the surface created when the original 2D curve revolves around the \(z\)-axis.

Important aspects of surface equations include:

• They provide a framework for understanding the geometric properties of surfaces
• Equations can be manipulated to reveal different characteristics like symmetry and curvature
• Cylindrical forms are particularly suited for problems involving rotations or circular symmetry

The equation displays how the radius \(r\) varies with \(z\), indicating that as \(z\) changes, the radius adjusts according to the parabolic curve rooted in the original hyperbola's revolution.