Problem 28 - Essential Calculus Early Transcendentals

Question

\(28-30=\) Find a vector function that represents the curve of intersection of the two surfaces. The cylinder \(x^{2}+y^{2}=4\) and the surface \(z=x y\)

Step-by-Step Solution

Answer

The vector function is \( \mathbf{r}(t) = \langle 2 \cos(t), 2 \sin(t), 2 \sin(2t) \rangle \).

Step 1: Re-parameterize the Cylinder Equation

The cylinder equation is given by \(x^{2} + y^{2} = 4\). We can parameterize this circle in the \(xy\)-plane by expressing \(x\) and \(y\) using trigonometric functions: \(x = 2 \cos(t)\) and \(y = 2 \sin(t)\). This parameterization ensures that \(x^{2} + y^{2} = 4\) as needed.

Step 2: Substitute Parameterization into Surface Equation

Given the surface equation \(z = xy\), substitute our parameterized values for \(x\) and \(y\). Therefore, \(z = (2 \cos(t))(2 \sin(t)) = 4 \cos(t) \sin(t)\), which can be rewritten using a trigonometric identity as \(z = 2 \sin(2t)\).

Step 3: Construct the Vector Function

We combine the parameterization of \(x\), \(y\), and \(z\) to formulate the vector function. The vector function is given by \(\mathbf{r}(t) = \langle 2 \cos(t), 2 \sin(t), 2 \sin(2t) \rangle\), where \(t\) is the parameter.

Step 4: Verify the Solution

To verify, check that the vector function satisfies both original equations. For \(x\) and \(y\): \((2 \cos(t))^2 + (2 \sin(t))^2 = 4\). For \(z\): \(z = xy = 2 \cos(t) \times 2 \sin(t) = 4 \cos(t) \sin(t) = 2 \sin(2t)\). Both conditions are satisfied, confirming that \( \mathbf{r}(t) \) correctly represents the curve.

Key Concepts

Parameterization

Parameterization is a method that helps to express complex geometric objects, like curves or surfaces, using simpler mathematical expressions called parameters. When given an equation such as a cylinder, parameterization simplifies by using trigonometric functions to represent coordinates.
For the cylinder described by the equation \(x^2 + y^2 = 4\), the goal is to find a way to represent \(x\) and \(y\) so that they always satisfy this equation. By using parameter \(t\), we can set \(x = 2 \cos(t)\) and \(y = 2 \sin(t)\).
This ensures that \(x^2 + y^2\) equals 4 at any time because of the trigonometric identity \(\cos^2(t) + \sin^2(t) = 1\). When multiplied by the radius squared (4, since the radius is 2), this identity holds true. Parameterization simplifies working with such equations on the plane and in space.

Curve Intersection

In mathematics, curve intersection is about finding the set of points that are shared by two curves. This forms the intersection of two geometric surfaces or paths.
Here, we are tasked with finding where the cylinder \(x^2 + y^2 = 4\) intersects with the surface \(z = xy\). The method involves substituting the parameters used for the cylinder directly into the surface equation.
By taking the parameterized \(x\) and \(y\) values \(x = 2 \cos(t)\) and \(y = 2 \sin(t)\), and substituting them into the equation \(z = xy\), we derive \(z = 4 \cos(t) \sin(t)\). This can be rewritten using a trigonometric identity: \(z = 2 \sin(2t)\). This transformation demonstrates the intersection as it curves through three-dimensional space, creating a clear relationship between the two surfaces. It translates the geometric complexity into algebraic simplicity.

Cylinder Equation

The cylinder equation in this context refers to a mathematical description of a cylinder in three-dimensional space. Specifically, the equation \(x^2 + y^2 = 4\) describes a vertical cylinder centered around the \(z\)-axis with a radius of 2.
The surface is independent of \(z\), which means it extends infinitely in both the positive and negative \(z\)-axis directions.
This equation expresses a basic geometric shape in terms of algebra, highlighting the coordinate system basis which is the \(xy\)-plane. It simplifies working with curves and surfaces by reducing them to manageable equations that align with familiar geometric principles.
The simplicity of the cylinder equation emerges from its unchanging radius and a consistent center point, making it a crucial component when investigating intersections and parameterizations.