Problem 6 - Thomas Calculus

Question

In Exercises \(1-12,\) sketch the graph described by the following cylindrical coordinates in three-dimensional space. $$ z=r \sin \theta $$

Step-by-Step Solution

Answer

The graph is a plane in 3D where \(z = y\).

Step 1: Understanding the Equation

The given equation is in cylindrical coordinates: \[ z = r \sin \theta \] Here, \(z\) represents the height, \(r\) is the radial distance, and \(\theta\) is the angle in the \(xy\)-plane.

Step 2: Rewriting in Cartesian Coordinates

Convert the cylindrical equation into Cartesian coordinates using the relations:- \(x = r \cos \theta\)- \(y = r \sin \theta\)We know that \(r = \sqrt{x^2 + y^2}\). Substitute \(r\) in the expression:\[ z = (\sqrt{x^2 + y^2}) \sin \theta \]Now, using \(y = r \sin \theta\), we find that\[ z = y \]

Step 3: Graphing the Equation

The expression \(z = y\) in Cartesian coordinates describes a plane in three-dimensional space. This means for any point, the z-coordinate is equal to the y-coordinate. The plane passes through every point in space where \(z = y\), cutting the xy-plane along the line \(y = x\).

Step 4: Finalizing the Graph

Plot the plane in three dimensions that slices through the center, where height \(z\) equals the distance \(y\). It is a flat sheet, with no curvature, extending infinitely in the x-direction.

Key Concepts

3D Graphing

Graphing in three dimensions adds an exciting depth to visualizing mathematical equations. Unlike flat, two-dimensional graphing, 3D graphing introduces an additional axis, often depicted as the z-axis. This axis allows us to represent height or depth, bringing a more comprehensive spatial understanding. Imagine plotting points not just along a plane, but also rising or sinking into space.
To graph equations like the one given, \[z = r \sin \theta\],we rely on understanding how these points relate spatially. In this case, the graph results in a plane where the z-value matches the corresponding y-value, forming a diagonal slice through the space.
**Key Points to Remember:**

• Three axes typically labeled x, y, and z.
• Enables the visualization of complex surfaces or planes.
• Each point is represented by an (x, y, z) coordinate.

Cartesian Coordinates Conversion

Converting between coordinate systems is a crucial skill for understanding equations in different contexts. In our problem, we start with cylindrical coordinates and convert to Cartesian coordinates to make interpreting the plane easier.
**Cylindrical to Cartesian Conversion Rules:**

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

Using these conversions, we simplified the original equation \(z = r \sin \theta\) to \(z = y\), helping us graph and understand the plane within the familiar Cartesian framework.
This conversion offers a clearer geometric interpretation, facilitating easier graphing and analysis.

Plane Equation

A plane equation in three-dimensional space describes a flat, two-dimensional surface stretching infinitely. For instance, in our exercise, the converted Cartesian equation \(z = y\) represents such a plane.
**Understanding Plane Equations:**

• The equation \(z = y\) indicates every point above or below the xy-plane where the z-value directly equals the y-value.
• This creates a diagonal plane slicing through the space, cutting across the line \(y = x\) in the xy-plane.
• Unlike curved surfaces, planes have a uniform behavior with no curvature, extending in all directions along with their defined inclinations.

Understanding these properties allows for a more profound appreciation of spatial representations in mathematical problems.

Radial Distance

Radial distance is a key concept in cylindrical coordinates, where it measures how far a point is from the origin in the plane. Think of it as the radius of a circle centered at the origin.
**Important Notes about Radial Distance:**

• Represented by \(r\), it is calculated as \(r = \sqrt{x^2 + y^2}\).
• It provides a scalable measure that captures distances in the plane from the center outward.
• In relation to the exercise, it helped transition from the cylindrical equation \(z = r \sin \theta\) to a simpler Cartesian form.

Comprehending radial distance is indispensable when working with cylindrical coordinates, as it assists in understanding transformations and relationships between equations.