Problem 14 - Thomas Calculus

Question

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.) Tilted plane inside cylinder The portion of the plane \(x-y+2 z=2\) $$\begin{array}{l}{\text { a. Inside the cylinder } x^{2}+z^{2}=3} \\ {\text { b. Inside the cylinder } y^{2}+z^{2}=2}\end{array}$$

Step-by-Step Solution

Answer

Parametrize using ellipse formulas for cylinders and solve for remaining variables: (a) use \(x = \sqrt{3} \cos(t)\) and (b) use \(y = \sqrt{2} \cos(s)\).

Step 1: Understand the Problem

The problem is to find a parametrization for portions of a plane within given cylinders. For part (a), the portion of the plane is inside a cylinder with equation \(x^2+z^2=3\). For part (b), the plane portion is inside a cylinder with equation \(y^2+z^2=2\). First, identify how the plane can be expressed in parametric form.

Step 2: Parametrize the Plane

The given plane is \(x - y + 2z = 2\). Rearrange this to express one variable in terms of the others, such as \(x = y - 2z + 2\). Use parameters \(y\) and \(z\) to define the variables in the plane: \(x = y - 2z + 2,\) \(y = y,\) \(z = z\). A general parametrization of the plane could be \((y - 2z + 2, y, z)\) with parameters \(y\) and \(z\).

Step 3: Parametrize Inside the Cylinder for Part (a)

For the cylinder \(x^2 + z^2 = 3\), use a circular parametrization. Let \(x = \sqrt{3} \cos(t)\) and \(z = \sqrt{3} \sin(t)\), where \(t\) is the parameter ranging from \(0\) to \(2\pi\). Substitute \(x = \sqrt{3} \cos(t)\) into the plane's equation: \(\sqrt{3}\cos(t) - y + 2\sqrt{3} \sin(t) = 2\), solve for \(y\): \(y = \sqrt{3}\cos(t) + 2\sqrt{3}\sin(t) - 2\). The parametrization is \((\sqrt{3}\cos(t), \sqrt{3}\cos(t) + 2\sqrt{3}\sin(t) - 2, \sqrt{3}\sin(t))\).

Step 4: Parametrize Inside the Cylinder for Part (b)

For the cylinder \(y^2 + z^2 = 2\), use a circular parametrization. Let \(y = \sqrt{2} \cos(s)\) and \(z = \sqrt{2} \sin(s)\), where \(s\) is a parameter from \(0\) to \(2\pi\). Substitute \(y = \sqrt{2} \cos(s)\) into the plane's equation: \(x - \sqrt{2}\cos(s) + 2\sqrt{2}\sin(s) = 2\), solve for \(x\): \(x = \sqrt{2}\cos(s) - 2\sqrt{2}\sin(s) + 2\). The parametrization becomes \((\sqrt{2}\cos(s) - 2\sqrt{2}\sin(s) + 2, \sqrt{2}\cos(s), \sqrt{2}\sin(s))\).

Key Concepts

Cylindrical Coordinates

To understand how the surface inside the given cylinders is parametrized, we need to first explore cylindrical coordinates. This is a coordinate system that extends the concept of polar coordinates to three dimensions. Here are the components of cylindrical coordinates:

• The radius from the origin in the xy-plane, denoted as \(r\).
• The angle \(\theta\) around the z-axis, similar to polar coordinates in 2D.
• The height \(z\), which is the same as in Cartesian coordinates.

By using cylindrical coordinates, we can easily describe cylinders and surfaces around the z-axis. For example, the equation \(x^2 + z^2 = 3\) can be rewritten in cylindrical form as \(r^2 = 3\), or \(r = \sqrt{3}\). This makes it simpler to parametrize circular paths or areas within the cylinder.
In these exercises, parametrizations use variations of these equations to define surfaces constrained within specified cylinders, leveraging the ease of description in cylindrical coordinates.

Parametric Equations

A parametric equation represents a geometric object in a different, often more flexible format by using one or more parameters. When working with surfaces and curves, parametric equations help visualize and calculate points in a space without the direct use of standard x, y, z coordinates.
For the plane \(x - y + 2z = 2\), we chose \(y\) and \(z\) as parameters to express \(x\), resulting in the parametric form \((y - 2z + 2, y, z)\). This transformation allows us to map out the entire plane using variations in \(y\) and \(z\), essentially stitching together each point.
Within the cylinder constraints, the parametric equations employ trigonometric functions to map each outline, such as \(x = \sqrt{3} \cos(t)\) and \(z = \sqrt{3} \sin(t)\) for the cylinder \(x^2 + z^2 = 3\). Using these parametric forms, we can explore different mathematical entities and ensure we remain within the specified surfaces.

Calculus

In calculus, parametrizations can be particularly useful to simplify calculations involving curves and surfaces. They provide a way to handle complex shapes by transforming them into simpler parameter-based equations. This allows us to explore areas, lengths, and other properties using differential calculus.
When we parametrized the surface of the plane under the constraint of a cylinder, we essentially prepared the problem for further calculus operations, such as finding tangent planes, normal vectors, or evaluating integrals over the surface.

• The use of trigonometric functions like \(\cos\) and \(\sin\) in \(x = \sqrt{3}\cos(t)\) and \(z = \sqrt{3}\sin(t)\), helps in smoothly mapping out the surface.
• Integral calculus can then use this setup to calculate areas or volumes related to the surface, framed by these curves.

In essence, parametrization bridges geometric intuition and calculus technique, enabling us to analyze properties of complex surfaces with more ease and precision.