Problem 1 - Thomas Calculus

Question

In Exercises \(1-12,\) sketch the graph described by the following cylindrical coordinates in three-dimensional space. $$ r=2 $$

Step-by-Step Solution

Answer

The graph is a vertical cylinder of radius 2 centered on the z-axis.

Step 1: Understand the Cylindrical Coordinate System

Cylindrical coordinates consist of three components: \(r, \theta,\) and \(z\). Here, \(r\) represents the radial distance from the origin in the xy-plane, \(\theta\) is the angle measured from the positive x-axis in the counterclockwise direction, and \(z\) is the height above the xy-plane.

Step 2: Analyze the Given Parameter

The given problem specifies \(r = 2\). This means that in the xy-plane, any point that is 2 units away from the origin could potentially be part of the graph.

Step 3: Determine the Shape in the xy-plane

Since \(r = 2\) holds for any angle \(\theta\), the points in the xy-plane form a circle centered at the origin with a radius of 2 units.

Step 4: Extend into the Third Dimension

In cylindrical coordinates, the \(r = 2\) indicates a cylindrical surface stretching infinitely in the \(z\) direction since there is no restriction on \(\theta\) or \(z\). This results in a vertical cylinder with radius 2.

Step 5: Sketch the Graph

To sketch the graph, draw a circle of radius 2 in the xy-plane and then extend this circle uniformly in both positive and negative directions along the z-axis, forming a cylindrical surface.

Key Concepts

3D graph sketching

When learning to sketch 3D graphs, it's essential to grasp how each dimension impacts the overall shape. Imagine you're drawing a picture, not just taking notes. Start by focusing on one part of the graph, such as the xy-plane, and identify the fundamental shape. For cylindrical coordinates, this often involves circles or spirals.
Next, take that shape and gently extend it into the third dimension. This may mean transforming a circle into a cylinder, or a spiral into a helical structure. Always check that your sketch makes sense by rotating or changing your perspective. Use shading or dotted lines to indicate depth.

• Identify primary shapes (e.g., circles).
• Project these shapes into 3D (e.g., cylinders).
• Ensure your sketch represents three dimensions accurately using depth indicators.

By breaking down the complex 3D shapes, you can build more accurate and understandable sketches.

cylindrical surface

A cylindrical surface is a common structure in 3D space, defined by maintaining a constant radial distance, or radius, from the axis of the cylinder. Conceptually, imagine wrapping a sheet of paper around a straight line; this can simulate the creation of a cylindrical surface.
In cylindrical coordinates, a fixed "r" implies you're working with a cylinder rather than a single point or line. The angle \(\theta\) and height \(z\) can vary freely, leading to a surface extending infinitely in the direction parallel to the z-axis.

• Constant radial distance defines the surface.
• Infinite extension in the direction of the height or axis.
• Creating circles in the lateral view and a rectangle in the cross-section.

Understanding cylindrical surfaces is foundational in visualizing more complex 3D shapes.

radial distance

Radial distance, often symbolized as \(r\), represents how far a point is from the z-axis, which acts as the central line in cylindrical coordinates. Visualize radial distance as the length of the radius in the xy-plane extending from a point on the circle's edge to its center at the origin.
Radial distance is always non-negative, as it describes a magnitude and not a direction. By keeping the radial distance constant, you will find that the points all lie on a circle in the xy-plane. This circle, when extended along the z-axis, forms a cylindrical shape.

• Distance from the center to a point in the xy-plane.
• Always a positive value, depicting magnitude.
• Key to forming cylindrical shapes when extended upwards.

Grasping the concept of radial distance is crucial for comprehending how cylindrical coordinates create 3D structures.

cylinder in space

A cylinder in space is a three-dimensional shape with circular cross-sections and parallel sides. Using cylindrical coordinates makes it easy to define and understand cylinders, thanks to the fixed radial distance and the infinite extension in the height (z) direction.
To visualize this, imagine the tube of a paper towel that extends infinitely up and down without any end caps. The value of \(r\), or the radial distance, remains constant, creating a consistent circular cross-section regardless of height. \(\theta\), the angle from the x-axis, can change, indicating rotational symmetry around the central axis.

• Defined by maintaining a constant \(r\) value.
• Forms a surface due to the range of \(\theta\) and indefinite \(z\).
• Visualize as infinite in height, emphasizing its structural symmetry.

Recognizing how a cylinder behaves and can be interpreted in a spatial context aids in mastering its role in mathematical and physical applications.