Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

The part of the saddle \(a z=x^{2}-y^{2}\) inside the cylinder \(x^{2}+y^{2}=a^{2}, a>0\)

Step-by-Step Solution

Answer

The part of the saddle inside the cylinder is parameterized as \( (x, y, z) = (a \cos(\theta), a \sin(\theta), a \cos(2\theta)) \) with \( \theta \in [0, 2\pi] \).

Step 1: Identify the Surfaces

We have two surfaces given in the problem: the saddle-shaped surface \(z = \frac{1}{a}(x^2 - y^2)\) and the cylinder \(x^2 + y^2 = a^2\). We want the portion of the saddle that intersects with the cylinder.

Step 2: Express the Saddle Equation in terms of the Cylinder

Within the boundaries of the cylinder \(x^2 + y^2 = a^2\), we express the saddle surface as \(z = \frac{1}{a}(x^2 - y^2)\), which we will plot only for points that satisfy the cylindrical condition.

Step 3: Parameterize the Cylinder

Since the cylinder equation is \(x^2 + y^2 = a^2\), we convert it to polar coordinates: \(x = a \cos(\theta)\) and \(y = a \sin(\theta)\). Here \(\theta\) ranges from 0 to \(2\pi\) to cover the entire cylinder.

Step 4: Substitute Polar Coordinates into Saddle Equation

Substitute \(x = a \cos(\theta)\) and \(y = a \sin(\theta)\) into the saddle equation: \[z = \frac{1}{a}(a^2 \cos^2(\theta) - a^2 \sin^2(\theta)) = a \cos(2\theta)\]. So now, \(z = a \cos(2\theta)\).

Step 5: Define the Range of the Parameter

Since \(\theta\) ranges from 0 to \(2\pi\) for a full circle around the cylinder, we substitute this into our parameterized equation. Thus, the part of the saddle inside the cylinder is parameterized by \((x, y, z) = (a \cos(\theta), a \sin(\theta), a \cos(2\theta))\) where \(\theta \in [0, 2\pi]\).

Key Concepts

Parameterization

Parameterization is a way to describe a surface or shape using parameters, often for converting complex equations into simpler forms for analysis. Think of it as using angles and distances to describe a location instead of strict coordinates. By parameterizing, we simplify our work with surfaces, making it easier to study intersections or transformations. In this exercise, we parameterize the cylinder using an angle \(\theta\). This translates the fixed-form equation of the cylinder \(x^2 + y^2 = a^2\) into expressions using trigonometric functions: \(x = a \cos(\theta)\) and \(y = a \sin(\theta)\). This helps capture any point on the circular cross-section of the cylinder effectively.

Polar Coordinates

Polar coordinates are an alternative coordinate system to the familiar Cartesian coordinates (x, y). In polar coordinates, a point's location is determined by its distance from the origin, called the radius, and an angle from a reference direction. This is particularly useful for circular or cylindrical shapes.

• Radius or radial distance measures from the origin to the point.
• Angle (often denoted as \(\theta\)) measures the rotation from the positive x-axis.

In the problem, when we convert the cylinder's equation to polar form, we use polar coordinates to manipulate and analyze the geometry. The transformation \(x = a\cos(\theta)\) and \(y = a\sin(\theta)\) proves useful since solving problems involving circles or curves becomes easier in this system.

Saddle Surface

A saddle surface is a fascinating structure encountered frequently in calculus and differential geometry. It looks like a horse's saddle or a Pringles chip, with both concave and convex sections. The standard form, \(z = \frac{1}{a}(x^2 - y^2)\), creates a hyperbolic paraboloid, displaying unique behaviors:

• Along one axis, it curves upward, like a valley.
• Along the perpendicular axis, it curves downward, like a hill.

The task here examines the section of this saddle found within a cylinder. By expressing in terms of polar coordinates, the problem reframes the saddle as \(z = a\cos(2\theta)\). This cleverly simplifies the visualization of the surface inside the constraint.

Cylinder

Cylinders are common 3D shapes with a circular base and a specific height. Mathematically, for a cylinder aligned along an axis, we often use \(x^2 + y^2 = a^2\) to represent its cross-section or base circle. This exercise considers a vertical cylinder centered at the origin with a circular base. Understanding cylindrical geometry is crucial when dealing with circular motion or paths. They hold unique properties:

• The circular aspect simplifies with the use of radial and angular parameters (often in polar form).
• They provide constraints that can limit the volume or area calculations.

By parameterizing with angles, many problems related to intersections, like the problem here, become easier to solve.