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Chapter 29: Problem 35

Perform the indicated operations involving cylindrical coordinates. Describe the surface for which the cylindrical coordinate equation is (a) \(r=2,(\mathrm{b}) \theta=2,(\mathrm{c}) z=2\)

Short Answer

Expert verified
(a) Infinite vertical cylinder, (b) Half-plane at 2 radians, (c) Plane at height 2.

Step by step solution

01

Understand Cylindrical Coordinates

Cylindrical coordinates consist of three components: * \( r \): the radial distance from the z-axis.* \( \theta \): the angle in the \(xy\)-plane from the positive \(x\)-axis.* \( z \): the height above or below the \(xy\)-plane.
02

Describe the Surface for \( r = 2 \)

In cylindrical coordinates, fixing \( r = 2 \) implies a circular cylinder centered around the \( z \)-axis with a radius of 2. This means for every \( (x,y) \), \( x^2 + y^2 = 4 \). The height \( z \) can be any value, so the surface is an infinite vertical cylinder along the \(z\)-axis.
03

Describe the Surface for \( \theta = 2 \)

Fixing \( \theta = 2 \) means all points make an angle of 2 radians with the positive \(x\)-axis. This arises as a half-plane extending in the \(z\)-direction, where the points lie in a plane that slices through the \(z\)-axis at a specific angle (2 radians) with respect to the positive \(x\)-axis.
04

Describe the Surface for \( z = 2 \)

Fixing \( z = 2 \) creates a plane parallel to the \(xy\)-plane, but located at a height \( z = 2 \). This surface is infinite in the \(xy\)-direction and has no dependency on variants in \(r\) and \( \theta \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circular Cylinder
A circular cylinder in cylindrical coordinates is an intriguing geometrical concept. When we talk about a circular cylinder, we imagine an infinite three-dimensional shape that wraps symmetrically around an axis, much like a soup can. In the context of cylindrical coordinates:
  • "Circular" signifies that the cross-section intersecting this cylinder at any height is a circle.
  • "Cylinder" refers to its constant radius and its continuity along a linear axis, typically the z-axis.
When you fix the radial distance, such as setting \( r = 2 \), it means that every point on the surface of the cylinder is exactly 2 units away from the center line, or the z-axis. The equation \( x^2 + y^2 = 4 \) represents this circle on any horizontal plane at height \( z \). The value of \( z \) can vary, indicating that the cylinder stretches infinitely upwards and downwards along the z-axis.
Coordinate Plane
The coordinate plane in the context of cylindrical coordinates plays a pivotal role in understanding how points are organized spatially. Generally, a coordinate plane is a two-dimensional flat surface formed by intersecting the x and y axes at a right angle, also known as the \( xy \)-plane.
In cylindrical coordinates, the surface being described when fixing a certain variable, such as \( z \), results in variation or intersection with a plane:
  • For \( z = 2 \), the plane moves parallel to the \( xy \)-plane and sits at the height of 2 units, forming a horizontal surface.
  • This plane is infinite in the directions perpendicular to the z-axis, stretching along the x and y directions.
In cylindrical terms, this plane can be visualized distinctly from the \( xy \)-plane but follows the same principles of coordinate intersection and expansion along the x and y dimensions.
Radial Distance
The radial distance in cylindrical coordinates is a fundamental aspect that denotes the length from the z-axis to any point in the plane.
This is represented by the term \( r \) in the equations:
  • It's similar to the radius of a circle, indicating how far away a point is from the center axis.
  • By fixing \( r \), such as stating \( r = 2 \), a fixed circle is described in the \( xy \)-plane.
  • Alterations in \( r \) affect how points are perceived within the circular plane, impacting the entire shape or figure associated with the value.
Radial distance is critical because altering it redefines spatial relationships and can create infinite cylindrical shapes or distinct circular intersections within such coordinate systems.
Angles in Mathematics
Angles in mathematics serve as a critical foundation for plotting points and describing surfaces within the coordinate system.
  • In cylindrical coordinates, \( \theta \) describes the angle made with the positive x-axis in the \( xy \)-plane.
  • This angular measurement determines direction, a concept used to represent line orientation or plane orientation within a multidimensional space.
  • For example, an angle \( \theta = 2 \) implies that all points form a line making a 2 radian angle with the positive x-axis, often equating to creating a half-plane extending through the z-axis.
Such angles are integral for creating planes or delineating directions and can be flipped or rotated within the plane for different mathematical explorations or interpretations.

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Most popular questions from this chapter

The reciprocal of the image distance \(q\) from a lens as a function of the object distance \(p\) and the focal length \(f\) of the lens is \(\frac{1}{q}=\frac{1}{f}-\frac{1}{p} .\) Find the image distance of an object \(20 \mathrm{cm}\) from a lens whose focal length is \(5 \mathrm{cm}\).

Find the indicated volumes by double integration. The volume bounded by the planes \(x+3 y+2 z-6=0\), \(2 x=y, x=0,\) and \(z=0\)

Sketch the graphs of the given equations in the rectangular coordinate system in three dimensions. $$x y=2$$

Find the rectangular coordinates of the points whose cylindrical coordinates are (a) \((4, \pi / 3,2),(\mathrm{b})(5, \pi,-4)\)

Perform the indicated operations involving cylindrical coordinates. Write the equation \(r=2 \sin \theta\) in rectangular coordinates and sketch the surface.
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