Problem 59 - Calculus III

Question

Find a vector function that describes the following curves. Intersection of the cylinder \(x^{2}+y^{2}=4\) with the plane \(x+z=6\)

Step-by-Step Solution

Answer

The vector function describing the curve is \(\langle 2\cos(\theta), 2\sin(\theta), 6 - 2\cos(\theta) \rangle\).

Step 1: Express x and y in terms of a parameter

To describe the intersection curve of the cylinder \(x^2 + y^2 = 4\), we can express \(x\) and \(y\) using a parameter, such as \(\theta\). A common parametrization for a circle is to let \(x = 2\cos(\theta)\) and \(y = 2\sin(\theta)\).

Step 2: Compute z using the plane equation

Substitute the expression for \(x\) into the plane equation \(x + z = 6\). This gives us \(z = 6 - x = 6 - 2\cos(\theta)\).

Step 3: Formulate the vector function

Combine the parameterizations from steps 1 and 2 into a single vector function. The vector function \(\mathbf{r}(\theta)\) can be written as: \[ \mathbf{r}(\theta) = \langle 2\cos(\theta), 2\sin(\theta), 6 - 2\cos(\theta) \rangle \] where \(\theta\) ranges over an interval that describes the position on the circle.

Key Concepts

Understanding Parametrization

Parametrization is a way to define a set of points in space using one or more parameters. It is commonly used in mathematics and physics to describe curves and surfaces. Instead of expressing functions in terms of typical variables like \(x\) and \(y\), parametrization employs a parameter, often denoted as \(t\) or \(\theta\).

For example, in the exercise, we use the parameter \(\theta\) to describe a circle, which is part of the given intersection. The standard parametrization for a circle of radius \(r\) centered at the origin is \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). This transforms a circular shape into a different mathematical form that is easier to manipulate in vector functions.

Parametrization offers several benefits:

• It simplifies the computation of curves and their intersections.
• Provides a clear formula for calculating positions on curves.
• Makes it easier to convert equations into vector functions.

By understanding parametrization, you'll find it much easier to explore complex geometric shapes.

Cylindrical Coordinates

Cylindrical coordinates extend the idea of polar coordinates into three dimensions, making them ideal for dealing with problems involving cylindrical shapes. In cylindrical coordinates, any point in space is described using three variables: radius \(r\), angle \(\theta\), and height \(z\).

The relationships are as follows:

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

For example, the exercise involves a cylinder given by the equation \(x^2 + y^2 = 4\). Here, the radius \(r\) is 2, since \(r^2 = 4\). The height \(z\) is determined separately and can have any value initially, which is why cylindrical coordinates are so powerful: they allow for flexibility with height, while keeping the circular base intact.

Learning about cylindrical coordinates helps with understanding orientations and distances in space, especially when dealing with symmetrical shapes like cylinders.

Intersection of Surfaces

The concept of the intersection of surfaces involves finding a common curve or set of points shared by two or more surfaces. When surfaces intersect, they reveal unique curves that might not be easily seen through basic observation alone.

In the exercise, we find the intersection of a cylinder \(x^2 + y^2 = 4\) with a plane \(x + z = 6\). We use parametrization and substitution to discover this curve. By circumscribing a parameter into one surface's equation (in this case, the cylinder), we can substitute it into the other surface's equation, like the plane.

Once \(x\) is written as \(2\cos(\theta)\), substitute it into the plane equation to solve for \(z\), giving us a new expression: \(z = 6 - 2\cos(\theta)\). This completes the system of equations required for the vector function.

Intersection of surfaces is crucial in visualizing and manipulating shapes, as it gives insights into where and how different geometric objects meet. Understanding these concepts equips you with the knowledge needed to handle complex spatial relationships.