Problem 105 - Thomas Calculus

Question

Vertical planes in cylindrical coordinates a. Show that planes perpendicular to the \(x\) -axis have equations of the form \(r=a \sec \theta\) in cylindrical coordinates. b. Show that planes perpendicular to the \(y\) -axis have equations of the form \(r=b \csc \theta .\)

Step-by-Step Solution

Answer

a: Planes perpendicular to the x-axis have the form \(r = a \sec \theta\); b: Planes perpendicular to the y-axis have the form \(r = b \csc \theta\).

Step 1: Understand the Cartesian Plane Equation

To show that planes perpendicular to the x-axis have the form \(r = a \sec \theta\), start by understanding that such a plane in Cartesian coordinates can be defined as \(x = a\).

Step 2: Convert Cartesian Coordinates to Cylindrical Coordinates

In cylindrical coordinates, the relation between Cartesian coordinates \((x, y, z)\) and cylindrical coordinates \((r, \theta, z)\) is given by: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(z = z\).

Step 3: Derive Equation for Planes Perpendicular to x-axis

Substitute \(x = a\) in the equation \(x = r \cos \theta\), we get \(r \cos \theta = a\). Solve for \(r\) to obtain \(r = \frac{a}{\cos \theta}\), hence \(r = a \sec \theta\).

Step 4: Understand the Cartesian Plane Equation for y-axis

For planes perpendicular to the y-axis, the plane's equation in Cartesian coordinates is \(y = b\).

Step 5: Derive Equation for Planes Perpendicular to y-axis

In cylindrical coordinates, substitute \(y = b\) into \(y = r \sin \theta\). Solve for \(r\): \(r \sin \theta = b\) which gives \(r = \frac{b}{\sin \theta}\), or \(r = b \csc \theta\).

Key Concepts

Cartesian coordinates

Cartesian coordinates, also known as rectangular coordinates, represent points in space using three perpendicular axes: the x-axis, the y-axis, and the z-axis. In this system, any point can be specified by an ordered trio

• **x:** The distance along the horizontal axis (left-right direction).
• **y:** The distance along the vertical axis (up-down direction).
• **z:** The distance along the depth axis (front-back direction).

For example, a point at (2, 3, 1) has coordinates that signify it is 2 units along the x-axis, 3 units along the y-axis, and 1 unit along the z-axis.
This coordinate system is intuitive for simple calculations and is particularly useful for representing objects in 3D space.When dealing with geometrical problems in Cartesian coordinates, equations often appear in simple linear forms, like the equation of a plane. For instance, a plane perpendicular to the x-axis might be defined by the equation \(x = a\), where \(a\) is a constant indicating the plane's position.

perpendicular planes

Understanding perpendicular planes involves recognizing that these planes intersect a given axis at a 90-degree angle. For instance, a plane perpendicular to the x-axis can be visualized as a vertical surface extending infinitely in the y and z directions while intersecting the x-axis at a fixed point.
To express such a plane using Cartesian coordinates, one simply states \(x = a\). Similarly, a plane that is perpendicular to the y-axis is represented as \(y = b\).
These planes are important in geometry and vector calculus because they help define boundaries and surfaces within a 3D coordinate system.Switching to cylindrical coordinates, we translate these conditions into an alternate set that involves radial distance \(r\) and angular position \(\theta\). To denote a plane perpendicular to the x-axis, we use the form \(r = a \sec \theta\), extending vertically at a fixed distance 'a' in terms of its projection onto the x-axis. For the y-axis, this becomes \(r = b \csc \theta\). Understanding these forms allows for more advanced spatial analysis, especially for objects with radial symmetry.

conversion between coordinate systems

The conversion between coordinate systems is a critical skill in both mathematics and physics. It allows a problem to be understood in various contexts and through different perspectives.
Cylindrical coordinates are particularly useful when dealing with circular or symmetrical subjects, as they include a radial distance \(r\), an angular coordinate \(\theta\), and an axial component \(z\).For a point in Cartesian coordinates

• **(x, y, z)**, the conversion to cylindrical coordinates is performed as follows:
• **Radial distance \(r\):** \(r = \sqrt{x^2 + y^2}\).
• **Angle \(\theta\):** \(\theta = \tan^{-1}(\frac{y}{x})\).
• **Height \(z\):** The z-coordinate remains the same, \(z = z\).

This conversion is particularly useful for simplifying the mathematics involved in analysing systems with rotational symmetry, such as turbines or circular plots of land. Each form of coordinates offers a clearer insight into different physical phenomena, thereby aiding in problem-solving and visualization.