Cylindrical and Spherical Coordinates

Slides:



Advertisements
Similar presentations
Polar Coordinates We Live on a Sphere.
Advertisements

The Cross Product of Two Vectors In Space
Chapter 8: Functions of Several Variables
Cylindrical and Spherical Coordinates
Chapter 12:Section6 Quadric Surfaces
Polar Coordinate System 11.3 – Polar Coordinates Used to plot and analyze equations of conics (circles, parabolas, ellipses, and hyperbolas. Another method.
Chapter 7: Vectors and the Geometry of Space
Chapter 7: Vectors and the Geometry of Space
16 MULTIPLE INTEGRALS.
H.Melikian/12001 Recognize and use the vocabulary of angles. Use degree measure. Use radian measure. Convert between degrees and radians. Draw angles in.
MULTIPLE INTEGRALS Triple Integrals in Spherical Coordinates In this section, we will learn how to: Convert rectangular coordinates to spherical.
For each point (x,y,z) in R3, the cylindrical coordinates (r,,z) are defined by the polar coordinates r and  (for x and y) together with z. Example Find.
17 VECTOR CALCULUS.
Chapter 15 – Multiple Integrals
Chapter 15 – Multiple Integrals
Analytic Geometry in Three Dimensions
Copyright © Cengage Learning. All rights reserved. 15 Multiple Integrals.
Vectors and the Geometry of Space 9. 2 Announcement Wednesday September 24, Test Chapter 9 (mostly )
CHAPTER 14 Vectors in three space Team 6: Bhanu Kuncharam Tony Rocha-Valadez Wei Lu.
Let’s start with a little problem…
11 Analytic Geometry in Three Dimensions
15.9 Triple Integrals in Spherical Coordinates
TRIPLE INTEGRALS IN SPHERICAL COORDINATES
Chapter 7: The Cross Product of Two Vectors In Space Section 7.4 Written by Dr. Julia Arnold Associate Professor of Mathematics Tidewater Community College,
Multiple Integration 14 Copyright © Cengage Learning. All rights reserved.
Chapter 7: Vectors and the Geometry of Space
Vectors and the Geometry of Space
EMLAB 1 Chapter 1. Vector analysis. EMLAB 2 Mathematics -Glossary Scalar : a quantity defined by one number (eg. Temperature, mass, density, voltage,...
6.3 Polar Coordinates Day One
Copyright © Cengage Learning. All rights reserved. 0 Precalculus Review.
Warm-up Problems Sketch the surface 9x 2 + 4y 2 – 36z 2 – 18x – 144z = 171.
SECTION 12.6 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES.
11.7 Cylindrical and Spherical Coordinates. The Cylindrical Coordinate System In a cylindrical coordinate system, a point P in space is represented by.
Section 17.5 Parameterized Surfaces
Chapter 8: Functions of Several Variables Section 8.4 Differentials Written by Richard Gill Associate Professor of Mathematics Tidewater Community College,
Slide 5- 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Vectors and the Geometry of Space 9. Three-Dimensional Coordinate Systems 9.1.
Multiple Integration Copyright © Cengage Learning. All rights reserved.
Multiple Integrals 12.
Copyright © Cengage Learning. All rights reserved. 15 Multiple Integrals.
Polar Coordinates and Graphing
Chapter Nine Vectors and the Geometry of Space. Section 9.1 Three-Dimensional Coordinate Systems Goals Goals Become familiar with three-dimensional rectangular.
Transformations Dr. Hugh Blanton ENTC Dr. Blanton - ENTC Coordinate Transformations 2 / 29 It is important to compare the units that are.
ConcepTest Section 12.2 Question 1 Let be the distance above the ground (in feet) of a jump rope x feet from one end after t seconds. The two people turning.
Copyright © Cengage Learning. All rights reserved. 16 Vector Calculus.
Polygons and Solids. Helix - Volume or solid of spiral shape that turns at a constant angle. cylinder -Volume or solid generated by the rotation.
8.1 The Rectangular Coordinate System and Circles Part 2: Circles.
ConcepTest Section 19.2 Question 1 (a) The rectangle is twice as wide in the x-direction, with new corners at the origin, (2, 0, 0), (2, 1, 3), (0, 1,
Partial Derivatives Written by Dr. Julia Arnold Professor of Mathematics Tidewater Community College, Norfolk Campus, Norfolk, VA With Assistance from.
COORDINATE SYSTEMS & TRANSFORMATION
11.0 Analytic Geometry & Circles
Copyright © Cengage Learning. All rights reserved.
Cylindrical and Spherical Coordinates
11 Vectors and the Geometry of Space
Chapter 12 Math 181.
Copyright © Cengage Learning. All rights reserved.
Engineering Geometry Engineering geometry is the basic geometric elements and forms used in engineering design. Engineering and technical graphics are.
rectangular coordinate system spherical coordinate system
Copyright © Cengage Learning. All rights reserved.
Vectors and the Geometry of Space
Copyright © Cengage Learning. All rights reserved.
Chapter 8: Functions of Several Variables
Copyright © Cengage Learning. All rights reserved.
11 Vectors and the Geometry of Space
POLAR COORDINATES Dr. Shildneck.
Copyright © Cengage Learning. All rights reserved.
Cylindrical and Spherical Coordinates
15.7 Triple Integrals.
Copyright © Cengage Learning. All rights reserved.
Presentation transcript:

Cylindrical and Spherical Coordinates Written by Dr. Julia Arnold Associate Professor of Mathematics Tidewater Community College, Norfolk Campus, Norfolk, VA With Assistance from a VCCS LearningWare Grant

In this lesson you will learn about cylindrical and spherical coordinates how to change from rectangular coordinates to cylindrical coordinates or spherical coordinates how to change from spherical coordinates to rectangular coordinates or cylindrical coordinates how to change from cylindrical coordinates to rectangular coordinates or spherical coordinates

Polar Coordinates The polar coordinates r (the radial coordinate) and (the angular coordinate, often called the polar angle) are defined in terms of Cartesian Coordinates by where r is the radial distance from the origin, and is the counterclockwise angle from the x-axis. In terms of x and y,

Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height (z) axis. A point P is represented by an ordered triple of . As you can see, this coordinate system lends itself well to cylindrical figures. To change from rectangular to cylindrical: To change from cylindrical to rectangular:

 

Common Uses The most common use of cylindrical coordinates is to give the equation of a surface of revolution. If the z-axis is taken as the axis of revolution, then the equation will not involve theta at all. Examples: A paraboloid of revolution might have equation z = r2. This is the surface you would get by rotating the parabola z = x2 in the xz-plane about the z-axis. The Cartesian coordinate equation of the paraboloid of revolution would be z = x2 + y2. A right circular cylinder of radius a whose axis is the z-axis has equation r = R. A a sphere with center at the origin and radius R will have equation r + z2 = R2. A right circular cone with vertex at the origin and axis the z-axis has equation z = m r. As another kind of example, a helix has the following equations: r = R, z = a theta. http://mathforum.org/dr.math/faq/formulas/faq.cylindrical.html

Express the point (x,y,z) = (1, ,2) in cylindrical coordinates. Solution: Work it out before you go to the next slide.

Express the point (x,y,z) = (1, ,2) in cylindrical coordinates. Solution: You have two choices for r and infinitely many choices for theta. Thus the point can be represented by non unique cylindrical coordinates. For example See picture on next slide.

This graph was done using Win Plot in the two different coordinate systems.

The animation at the link below shows the points represented by constant values of the first coordinate as it varies from zero to one. http://www.math.montana.edu/frankw/ccp/multiworld/multipleIVP/cylindrical/body.htm                           The animation below the one above shows the points represented by constant values of the second coordinate as it varies from zero to 2 pi. You can also view at this link: http://www.tcc.edu/faculty/webpages/JArnold/movies.htm

Example 2  Identify the surface for each of the following equations. (a) r = 5 (b) (c) z = r   Solution: a. In polar coordinates we know that r = 5 would be a circle of radius 5 units. By adding the z dimension and allowing z to vary we create a cylinder of radius 5. 5

Example 2  Identify the surface for each of the following equations. (a) r = 5 (b) (c) z = r   Solution: b. This is equivalent to which we know to be a sphere centered at the origin with a radius of 10. 10 10 10

Example 2  Identify the surface for each of the following equations. (a) r = 5 (b) (c) z = r   Solution: c. Since the radius equals the height and the angle is any angle we get a cone.

Spherical coordinates are a system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid. The ordered triple is: For a given point P in spherical coordinates is the distance between P and the origin is the same angle theta used in cylindrical coordinates for is the angle between the positive z-axis and the line segment P (x,y,z) z O The figure at right shows the Rectangular coordinates (x,y,z) and The spherical coordinates

Conversion Formulas: Spherical to Rectangular: Rectangular to Spherical: Spherical to cylindrical ( ): Cylindrical to spherical ( ):

Example 3 A. Find a rectangular equation for the graph represented by the cylindrical equation B. Find an equation in spherical coordinates for the surface represented by each of the rectangular equations and identify the graph. 1. 2. Answers follow

Example 3 A. Find a rectangular equation for the graph represented by the cylindrical equation

Example 3 B. Find an equation in spherical coordinates for the surface represented by each of the rectangular equations and identify the graph. 1. A double cone.

Example 3 B. Find an equation in spherical coordinates for the surface represented by each of the rectangular equations and identify the graph. 2. A sphere

Review Rectangular to cylindrical: Cylindrical to rectangular: Spherical to Rectangular: Rectangular to Spherical: Spherical to cylindrical ( ): Cylindrical to spherical ( ):

For comments on this presentation you may email the author Dr. Julia Arnold at jarnold@tcc.edu.