Problem 13 - Thomas Calculus

Question

In Exercises \(1-16,\) find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) $$ \begin{array}{l}{\text { Tilted plane inside cylinder } \quad \text { The portion of the plane } x+y+} \\ {z=1}\end{array} $$ $$ \begin{array}{l}{\text { a. Inside the cylinder } x^{2}+y^{2}=9} \\ {\text { b. Inside the cylinder } y^{2}+z^{2}=9}\end{array} $$

Step-by-Step Solution

Answer

Parametrization inside cylinder \( x^2+y^2=9 \) is \( \mathbf{r}(\theta) = (3\cos\theta, 3\sin\theta, 1 - 3\cos\theta - 3\sin\theta) \); inside \( y^2+z^2=9 \) is \( \mathbf{r}(\theta) = (1 - 3\cos\theta - 3\sin\theta, 3\cos\theta, 3\sin\theta) \).

Step 1: Understand the problem

We need to find a parametric representation for the part of the plane \( x+y+z=1 \) inside specified cylinders.

Step 2a: Parametrize plane inside cylinder \( x^2+y^2=9 \)

We know the plane equation: \( x + y + z = 1 \). Choose parameters for \( x \) and \( y \) to describe the region inside the cylinder. Let, for instance, \( x = 3\cos(\theta) \), \( y = 3\sin(\theta) \), with \( 0 \leq \theta < 2\pi \). Then solve for \( z \): \( z = 1 - x - y = 1 - 3\cos(\theta) - 3\sin(\theta) \). Thus, the parametrization is: \( \mathbf{r}(\theta) = (3\cos(\theta), 3\sin(\theta), 1 - 3\cos(\theta) - 3\sin(\theta)) \).

Step 2b: Parametrize plane inside cylinder \( y^2+z^2=9 \)

Use parameters for \( y \) and \( z \) to fit inside this cylinder. Let, for instance, \( y = 3\cos(\theta) \), \( z = 3\sin(\theta) \), with \( 0 \leq \theta < 2\pi \). Solve for \( x \) from the plane equation: \( x = 1 - y - z = 1 - 3\cos(\theta) - 3\sin(\theta) \). Thus, the parametrization is: \( \mathbf{r}(\theta) = (1 - 3\cos(\theta) - 3\sin(\theta), 3\cos(\theta), 3\sin(\theta)) \).

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates offer a way to describe the location of a point in three-dimensional space using two distances and an angle. In essence, this system is very much like a combination of polar coordinates and Cartesian coordinates.

• Distance from the z-axis (like the radius in polar coordinates)
• Angle around the z-axis (like the angle in polar coordinates)
• Height along the z-axis (like the z-coordinate in Cartesian coordinates)

For example, in our problem's context, we're using cylindrical coordinates to define points on the circular cross-section of a cylinder. Each circular cross-section is determined by a fixed radial distance from the origin (in this case, 3, as both cylinders share a radius of 3) and an angle \( \theta \) ranging from 0 to \( 2\pi \).
Here, the expression \( x = 3\cos(\theta) \) and \( y = 3\sin(\theta) \) represents a point on the circle formed by the slice of the cylinder with radius 3. This effectively sweeps the surface of the cylinder as \( \theta \) varies, providing an elegant way to define the geometry of a cylinder through parametric equations.

Plane Equation

The equation of a plane is a mathematical expression that defines all the points which, when plotted, form a flat surface in three-dimensional space. The standard equation for a plane is \( ax + by + cz = d \).
In our example, the plane is given by the equation \( x + y + z = 1 \). This particular plane is not parallel to any of the axes as all coefficients (of \(x\), \(y\), and \(z\)) are non-zero and equal to 1.
When solving problems requiring planes within specific limits, such as within a cylinder, we express one variable in terms of the others. For instance, in our problem:

• Using the cylinder condition \( x^2 + y^2 = 9 \), we express \( z \) from the plane equation and use parametric equations for \(x\) and \(y\).
• For the condition \( y^2 + z^2 = 9 \), we express \( x \) in terms of \( y \) and \( z \).

This flexibility allows us to utilize the geometric boundaries defined by both the plane and the cylinder to create a comprehensive parametric representation.

Parametric Representation

A parametric representation expresses a set of related geometric quantities using parameters. It's particularly useful when we want to represent a two-dimensional surface in a three-dimensional space.
For the given task, we use a parameter \( \theta \) to denote points all around the cylinder’s circular base.

• For the first cylinder defined by \( x^2 + y^2 = 9 \), we let \(x = 3\cos(\theta)\) and \(y = 3\sin(\theta)\), solving for \(z\) as \(z = 1 - 3\cos(\theta) - 3\sin(\theta)\).
• For the second cylinder defined by \( y^2 + z^2 = 9 \), we let \(y = 3\cos(\theta)\) and \(z = 3\sin(\theta)\), solving for \(x\) as \(x = 1 - 3\cos(\theta) - 3\sin(\theta)\).

This use of parameterization allows for more flexibility in expressing complex surfaces and is exceptionally helpful when visualizing or performing calculations over them.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, play a key role in parametric equations, especially in translating between linear and circular or spherical coordinates.
In cylindrical coordinates, sine and cosine help describe the circular aspects of geometry.

• Cosine (\(\cos(\theta)\)) captures horizontal projections of circular motion.
• Sine (\(\sin(\theta)\)) captures vertical projections.

In our problem:

• The trigonometric function \(\cos(\theta)\) handles the x or y-coordinates depending on which cylinder we’re working with (\(x = 3\cos(\theta)\) or \(y = 3\cos(\theta)\)).
• Similarly, \(\sin(\theta)\) is used for the complementary coordinate (\(y = 3\sin(\theta)\) or \(z = 3\sin(\theta)\)).

By cycling through values from 0 to \(2\pi\), these functions allow us to map out the entire circumference in the parameterization, creating a seamless conversion from angle to coordinate positioning.