# 1.7 Cylindrical and Spherical Coordinates

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the position of each point on the plane is determined by an angle and a distance. The distance is usually denoted r and the angle is usually denoted θ. Thus, in this coordinate system, the position of a point will be given by the ordered pair (r, θ). These are called the polar coordinates These two quantities r and θ, are determined as follows. First, we need some reference points. You may recall that in the Cartesian coordinate system, everything was measured with respect to the coordinate axes. In the polar coordinate system, everything is measured with respect a fixed point called the pole and an axis called the polar axis. The is the equivalent of the origin in the Cartesian coordinate system. The polar axis corresponds to the positive x-axis. Given a point P in the plane, we draw a line from the pole to P. The distance from the pole to P is r, the angle, measured counterclockwise, by which the polar axis has to be rotated in order to go through P is θ. The polar coordinates of P are then (r, θ). Figure shows two points and their representation in the polar coordinate system. Remark 79 Let us make several remarks. 1. Recall that a positive value of θ means that we are moving counterclockwise. But θ can also be negative. A negative value of θ means that the polar axis is rotated clockwise to intersect with P. Thus, the same point can have several polar coordinates. For example, (, 90) and (, 70) represent the same point.. Recall that given an angle θ, the angles θ + π, θ + 4π,..., θ + kπ where k = 0, 1,, 3,... represent the same angle. Thus, (r, θ) and (r, θ + kπ) where k = 0, 1,, 3,... represent the same point.

2 1.7. CYLINDRICAL AND SPHERICAL COORDINATES Recall that a positive value of r means that the point is away from the pole in the direction of the positive x-axis (taking into account the rotation by θ). But r can also have a negative value. A negative value of r means that the point is away from the pole in the direction of the negative x-axis. Thus (r, θ) and ( r, θ + π) represent the same point. 4. The pole has infinitely many polar coordinates. They are of the form (0, θ) where θ can be anything. Why do we need another coordinate system? Cartesian coordinates are not best suited for every shape. Certain shape, including circular ones, cannot even be represented as a function in Cartesian coordinates. These shapes are more easily represented in polar coordinates. We will see some examples below. First, we need to see how to go back and forth between Cartesian and polar coordinates. Figure helps us understand how the two coordinate systems are related. We list the formulas which allow the switch between the two coordinate systems as a proposition. Proposition 80 The following formulas can be used to switch between Cartesian and polar coordinates. 1. From polar to Cartesian coordinates: Given r and θ, we can find x and y by: x = r cos θ (1.16) y = r sin θ. From Cartesian to polar coordinates: Given x and y, we can find r

3 58 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE and θ by: x + y = r (1.17) tan θ = y x We now look at some examples of converting points between polar and Cartesian coordinates. ( Example 81 Convert 4, π ) in Cartesian coordinates. 3 Using formulas 1.16 we have: x = 4 cos π ( 3 = 4 1 ) = ( ) y = 4 sin π 3 3 = 4 = 3 So, in Cartesian coordinates, the point is (, 3 ). Example 8 Convert (1, 0) in polar coordinates. We don t need a formula here. This point is on the x-axis, one unit from the origin. Its polar coordinates are (1, 0) ( ) Example 83 Convert, in polar coordinates. Using formulas 1.17, we have: ( ) ( ) r = + = 1 This gives us and r = ±1 tan θ = x y = = 1 Thus θ = π 4 With a choice of θ = π, we see that we must have r = 1. So, in polar coordinates, the point is 1, π ) ( 4. 4

4 1.7. CYLINDRICAL AND SPHERICAL COORDINATES 59 We now look at how to convert the equation of known shapes between Cartesian and polar coordinates. Recall that when we write the equation of a shape, we are writing conditions that the coordinates of a point must satisfy in order to belong to that shape. In Cartesian coordinates, this relation is between x and y. In polar coordinates, the relation will be between r and θ. Of course, the equations of the shapes you know in Cartesian coordinates will look very different in polar coordinates. To perform this, we will use the relations in equations 1.16 and Example 84 What is the equation of the line y = x + 5 in polar coordinates. Rewriting in terms of r and θ, we have There is not much to simplify here. r sin θ = r cos θ + 5 Example 85 What is the equation of the horizontal line through (1, ) in polar coordinates? In Cartesian coordinates, it is y =. This gives us r sin θ = Example 86 What is the equation of the circle of radius centered at the origin in polar coordinates? In Cartesian coordinates, this equation is x + y = 4. This gives us or r =. r cos θ + r sin θ = 4 r ( sin θ + cos θ ) = r = 4 Example 87 What is in Cartesian coordinate θ = 5? θ is related to x and y via the tangent function. θ = tan 1 y. Hence, we have x y = tan θ = tan 5 x that is y = (tan 5) x This is the equation of the line through the origin with slope tan 5. Polar coordinate can be extended to the three-dimensional space different ways. We present two here: the cylindrical coordinate system and the spherical coordinate system.

5 60 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.36: Cylindrical Coordinates 1.7. Cylindrical Coordinates These are coordinates for a three-dimensional space. In this coordinate system, a point P is represented by the triple (r, θ, z) where r and θ are the polar coordinates of the projection of P onto the xy-plane and z has the same meaning as in Cartesian coordinates. Figure 1.36 shows a point in cylindrical coordinates. Switching between Cartesian coordinates and cylindrical coordinates will use the same formulas as when switching between polar and Cartesian coordinates. We illustrate this with a few examples. ( 1 Example 88 Convert, ) 3, 5 to cylindrical coordinates. We see that and r = x + y ( ) ( 1 3 = + = 1 ) tan θ = y x = 3

6 1.7. CYLINDRICAL AND SPHERICAL COORDINATES 61 Thus θ = π 3 and r = 1. Thus, the cylindrical coordinates are ( 1, π 3, 5). Example 89 What is the equation in cylindrical coordinates of the cone x + y = z. Replacing x + y by r, we obtain r = z which usually gives us r = ±z. Since z can be any real number, it is enough to write r = z. Thus, in cylindrical coordinates, this cone is z = r. Example 90 What is the equation in Cartesian coordinates of r = 3. What is it? Since r = 3, we see that r = 9 that is x + y = 9. This is a cylinder centered along the z-axis, of radius Spherical Coordinates This is a three-dimensional coordinate system. The coordinates of a point P are given by the ordered pair (ρ, θ, ϕ) where: ρ is the distance from the origin to P. We assume ρ 0. θ has the same meaning as in polar and cylindrical coordinates. There are no restrictions on θ. ϕ is the angle between the positive z-axis and the line from the origin to P. We restrict ϕ to 0 ϕ π. This is illustrated by figure Let us first derive the formulas to switch between Cartesian and spherical coordinates. Using the triangle one can make with the z-axis and the line from the origin through P, we have z = ρ cos ϕ (1.18) and Also r = ρ sin ϕ (1.19) z + r = ρ (1.0) Using the triangle one can make with the x-axis and the line from the origin to the projection of P on the xy-plane, we have x = r cos θ (1.1) y = r sin θ and also x + y = r (1.)

7 6 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.37: Spherical Coordinates Combining relations 1.18, 1.19 and 1.1 gives us x = ρ sin ϕ cos θ y = ρ sin ϕ sin θ z = ρ cos ϕ Combining relations 1.0 and 1. gives us x + y + z = ρ In conclusion, the following relations can be used to switch between Cartesian and spherical coordinates: x = ρ sin ϕ cos θ (1.3) y = ρ sin ϕ sin θ z = ρ cos ϕ ρ = x + y + z We illustrate this with some examples. Example 91 Convert ( 1, 1, ) from Cartesian to spherical. We can get ρ easily. ρ = ( 1) ( ) = 4. Thus, ρ =. From

8 1.7. CYLINDRICAL AND SPHERICAL COORDINATES 63 z = ρ cos ϕ, we see that cosϕ = hence ϕ = cos 1 = 3 4π. Finally, to find θ, we can use the equation involving x or y. Let s use y. We have sin θ = y ρ sin ϕ = 1 = sin 1. Several values for θ would work. π 3π 4, 3π 4. 4 However, the projection of P on the xy-plane is in the second quadrant, so we must have θ = 3π 4. Thus, the spherical coordinates are (, 3π 4, ) 3π 4. Example 9 Convert ( 6, π 4, ) from cylindrical to spherical. Let us first note that θ is the same in both coordinates systems. Next, we can get ρ from equation 1.0. Since r = 6 and z =, we see that ρ = z + r = 8. Thus, ρ =. Finally, we use z = ρ cos ϕ to get ϕ. cosϕ = z ρ = = 1. Thus, ϕ = π 3. Hence, the spherical coordinates are (, π 4, π 3 ). Example 93 Identify the surface given by ρ =. Converting to Cartesian coordinates gives x +y +z = 4. Hence, it is a sphere centered at the origin, with radius. This is important to remember. In general, the equation ρ = c is the equation of the sphere of radius c centered at the origin in spherical coordinates. Make sure you can do the above before attempting the problems Problems Do odd # 3-3 at the end of 9.7 in your book.

### POLAR COORDINATES DEFINITION OF POLAR COORDINATES

POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand

### Section 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates

Section.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O,the rotating ray or half line from O with unit tick. A point P in

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### Example 1. Example 1 Plot the points whose polar coordinates are given by

Polar Co-ordinates A polar coordinate system, gives the co-ordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points

### Trigonometric Functions and Triangles

Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between

### Module 1 : A Crash Course in Vectors Lecture 2 : Coordinate Systems

Module 1 : A Crash Course in Vectors Lecture 2 : Coordinate Systems Objectives In this lecture you will learn the following Define different coordinate systems like spherical polar and cylindrical coordinates

### 2 Topics in 3D Geometry

2 Topics in 3D Geometry In two dimensional space, we can graph curves and lines. In three dimensional space, there is so much extra space that we can graph planes and surfaces in addition to lines and

### Chapter 5: Trigonometric Functions of Real Numbers

Chapter 5: Trigonometric Functions of Real Numbers 5.1 The Unit Circle The unit circle is the circle of radius 1 centered at the origin. Its equation is x + y = 1 Example: The point P (x, 1 ) is on the

### Midterm Exam I, Calculus III, Sample A

Midterm Exam I, Calculus III, Sample A 1. (1 points) Show that the 4 points P 1 = (,, ), P = (, 3, ), P 3 = (1, 1, 1), P 4 = (1, 4, 1) are coplanar (they lie on the same plane), and find the equation of

### Chapter 6 Trigonometric Functions of Angles

6.1 Angle Measure Chapter 6 Trigonometric Functions of Angles In Chapter 5, we looked at trig functions in terms of real numbers t, as determined by the coordinates of the terminal point on the unit circle.

### Trigonometric Functions: The Unit Circle

Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry

### Applications of Integration to Geometry

Applications of Integration to Geometry Volumes of Revolution We can create a solid having circular cross-sections by revolving regions in the plane along a line, giving a solid of revolution. Note that

### Mathematical Procedures

CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

### f = difference/major axis = ~1/300 for earth

Map Projection and Coordinates The shape of the earth Models Sphere with radius of ~6378 km Ellipsoid (or Spheroid) with equatorial radius (semi-major axis) of ~6378 km and polar radius (semi-minor axis)

### Chapter 12. The Straight Line

302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic- geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,

### 0.1 Linear Transformations

.1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Notation: f : A B If the value b B is assigned to value a A, then write f(a) = b, b is called

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### Area and Arc Length in Polar Coordinates

Area and Arc Length in Polar Coordinates The Cartesian Coordinate System (rectangular coordinates) is not always the most convenient way to describe points, or relations in the plane. There are certainly

### Two vectors are equal if they have the same length and direction. They do not

Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must

### This function is symmetric with respect to the y-axis, so I will let - /2 /2 and multiply the area by 2.

INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,

### Figure 1: u2 (x,y) D u 1 (x,y) yρ(x, y, z)dv. xρ(x, y, z)dv. M yz = E

Contiune on.7 Triple Integrals Figure 1: [ ] u2 (x,y) f(x, y, z)dv = f(x, y, z)dz da u 1 (x,y) Applications of Triple Integrals Let be a solid region with a density function ρ(x, y, z). Volume: V () =

### Number Sense and Operations

Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

### Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

### SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.

(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much

### Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible

### 13.4 THE CROSS PRODUCT

710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product

### Paths Between Points on Earth: Great Circles, Geodesics, and Useful Projections

Paths Between Points on Earth: Great Circles, Geodesics, and Useful Projections I. Historical and Common Navigation Methods There are an infinite number of paths between two points on the earth. For navigation

### POLAR COORDINATES: WHAT THEY ARE AND HOW TO USE THEM

POLAR COORDINATES: WHAT THEY ARE AND HOW TO USE THEM HEMANT D. TAGARE. Introduction. This note is about polar coordinates. I want to explain what they are and how to use them. Many different coordinate

### What You ll See in This Chapter. Word Cloud. Polar Coordinate Space. Fletcher Dunn. Ian Parberry University of North Texas

What You ll See in This Chapter Chapter 7 Polar Coordinate Systems Fletcher Dunn Valve Software Ian Parberry University of North Texas 3D Math Primer for Graphics & Game Development This chapter describes

### Triple Integrals in Cylindrical or Spherical Coordinates

Triple Integrals in Clindrical or Spherical Coordinates. Find the volume of the solid ball 2 + 2 + 2. Solution. Let be the ball. We know b #a of the worksheet Triple Integrals that the volume of is given

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

### Solutions to Homework 10

Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

### ORIENTATIONS OF LINES AND PLANES IN SPACE

GG303 Lab 1 9/10/03 1 ORIENTATIONS OF LINES AND PLANES IN SPACE I Main Topics A Definitions of points, lines, and planes B Geologic methods for describing lines and planes C Attitude symbols for geologic

### Section 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50

Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 5-37, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall

### Figuring out the amplitude of the sun an explanation of what the spreadsheet is doing.

Figuring out the amplitude of the sun an explanation of what the spreadsheet is doing. The amplitude of the sun (at sunrise, say) is the direction you look to see the sun come up. If it s rising exactly

### Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

### The Parabola and the Circle

The Parabola and the Circle The following are several terms and definitions to aid in the understanding of parabolas. 1.) Parabola - A parabola is the set of all points (h, k) that are equidistant from

### The Polar Coordinate System

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-008 The Polar Coordinate System Alisa Favinger University

### THE COMPLEX EXPONENTIAL FUNCTION

Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula

### Week 13 Trigonometric Form of Complex Numbers

Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working

### GRAPHING IN POLAR COORDINATES SYMMETRY

GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,

### Section 16.7 Triple Integrals in Cylindrical Coordinates

Section 6.7 Triple Integrals in Cylindrical Coordinates Integrating Functions in Different Coordinate Systems In Section 6.4, we used the polar coordinate system to help integrate functions over circular

### 88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

### Lines and planes in space (Sect. 12.5) Review: Lines on a plane. Lines in space (Today). Planes in space (Next class). Equation of a line

Lines and planes in space (Sect. 2.5) Lines in space (Toda). Review: Lines on a plane. The equations of lines in space: Vector equation. arametric equation. Distance from a point to a line. lanes in space

### Winter 2012 Math 255. Review Sheet for First Midterm Exam Solutions

Winter 1 Math 55 Review Sheet for First Midterm Exam Solutions 1. Describe the motion of a particle with position x,y): a) x = sin π t, y = cosπt, 1 t, and b) x = cost, y = tant, t π/4. a) Using the general

### 5.3 The Cross Product in R 3

53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

### Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

### Solutions to Vector Calculus Practice Problems

olutions to Vector alculus Practice Problems 1. Let be the region in determined by the inequalities x + y 4 and y x. Evaluate the following integral. sinx + y ) da Answer: The region looks like y y x x

### 6.3 Polar Coordinates

6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard

### AB2.5: Surfaces and Surface Integrals. Divergence Theorem of Gauss

AB2.5: urfaces and urface Integrals. Divergence heorem of Gauss epresentations of surfaces or epresentation of a surface as projections on the xy- and xz-planes, etc. are For example, z = f(x, y), x =

### Math Placement Test Practice Problems

Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211

### Electronic calculators may not be used. This text is not the text of solutions of exam papers! Here we will discuss the solutions of the exampapers.

EXAM FEEDBACK INTRODUCTION TO GEOMETRY (Math 20222). Spring 2014 ANSWER THREE OF THE FOUR QUESTIONS If four questions are answered credit will be given for the best three questions. Each question is worth

### 10 Polar Coordinates, Parametric Equations

Polar Coordinates, Parametric Equations ½¼º½ ÈÓÐ Ö ÓÓÖ Ò Ø Coordinate systems are tools that let us use algebraic methods to understand geometry While the rectangular (also called Cartesian) coordinates

### Evaluating trigonometric functions

MATH 1110 009-09-06 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,

### 1.5 Equations of Lines and Planes in 3-D

40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3-D Recall that given a point P = (a, b, c), one can draw a vector from

### 1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

### Math 1B, lecture 5: area and volume

Math B, lecture 5: area and volume Nathan Pflueger 6 September 2 Introduction This lecture and the next will be concerned with the computation of areas of regions in the plane, and volumes of regions in

### Chapter 6 Circular Motion

Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example

### Triangle Trigonometry and Circles

Math Objectives Students will understand that trigonometric functions of an angle do not depend on the size of the triangle within which the angle is contained, but rather on the ratios of the sides of

### Geometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi

Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in three-dimensional space, we also examine the

### Triple integrals in Cartesian coordinates (Sect. 15.4) Review: Triple integrals in arbitrary domains.

Triple integrals in Cartesian coordinates (Sect. 5.4 Review: Triple integrals in arbitrar domains. s: Changing the order of integration. The average value of a function in a region in space. Triple integrals

### Review B: Coordinate Systems

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of hysics 8.02 Review B: Coordinate Systems B.1 Cartesian Coordinates... B-2 B.1.1 Infinitesimal Line Element... B-4 B.1.2 Infinitesimal Area Element...

### Who uses this? Engineers can use angles measured in radians when designing machinery used to train astronauts. (See Example 4.)

1- The Unit Circle Objectives Convert angle measures between degrees and radians. Find the values of trigonometric functions on the unit circle. Vocabulary radian unit circle California Standards Preview

### Section 9.5: Equations of Lines and Planes

Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that

### Engineering Geometry

Engineering Geometry Objectives Describe the importance of engineering geometry in design process. Describe coordinate geometry and coordinate systems and apply them to CAD. Review the right-hand rule.

### MATH 275: Calculus III. Lecture Notes by Angel V. Kumchev

MATH 275: Calculus III Lecture Notes by Angel V. Kumchev Contents Preface.............................................. iii Lecture 1. Three-Dimensional Coordinate Systems..................... 1 Lecture

### MPE Review Section II: Trigonometry

MPE Review Section II: Trigonometry Review similar triangles, right triangles, and the definition of the sine, cosine and tangent functions of angles of a right triangle In particular, recall that the

### MA Lesson 19 Summer 2016 Angles and Trigonometric Functions

DEFINITIONS: An angle is defined as the set of points determined by two rays, or half-lines, l 1 and l having the same end point O. An angle can also be considered as two finite line segments with a common

### Math 115 Spring 2014 Written Homework 10-SOLUTIONS Due Friday, April 25

Math 115 Spring 014 Written Homework 10-SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain

### Pre-Calculus Review Problems Solutions

MATH 1110 (Lecture 00) August 0, 01 1 Algebra and Geometry Pre-Calculus Review Problems Solutions Problem 1. Give equations for the following lines in both point-slope and slope-intercept form. (a) The

### Figure 1: Volume between z = f(x, y) and the region R.

3. Double Integrals 3.. Volume of an enclosed region Consider the diagram in Figure. It shows a curve in two variables z f(x, y) that lies above some region on the xy-plane. How can we calculate the volume

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### 13 Groups of Isometries

13 Groups of Isometries For any nonempty set X, the set S X of all one-to-one mappings from X onto X is a group under the composition of mappings (Example 71(d)) In particular, if X happens to be the Euclidean

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155

Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate

### Trigonometry Notes Sarah Brewer Alabama School of Math and Science. Last Updated: 25 November 2011

Trigonometry Notes Sarah Brewer Alabama School of Math and Science Last Updated: 25 November 2011 6 Basic Trig Functions Defined as ratios of sides of a right triangle in relation to one of the acute angles

### Vik Dhillon, Sheffield University - UK

Spherical Trigonometry Vik Dhillon, Sheffield University - UK http://www.shef.ac.uk/uni/academic/n-q/phys/people/vdhillon/teaching/phy105/phy105_sphergeom.html INTRODUCTION Before we can understand the

### 9-3 Polar and Rectangular Forms of Equations

9-3 Polar and Rectangular Forms of Equations Find the rectangular coordinates for each point with the given polar coordinates Round to the nearest hundredth, if necessary are The rectangular coordinates

### 4.1: Angles and Radian Measure

4.1: Angles and Radian Measure An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other is called the terminal side. The endpoint that they share is

### Vector surface area Differentials in an OCS

Calculus and Coordinate systems EE 311 - Lecture 17 1. Calculus and coordinate systems 2. Cartesian system 3. Cylindrical system 4. Spherical system In electromagnetics, we will often need to perform integrals

### The Math Circle, Spring 2004

The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is Non-Euclidean Geometry? Most geometries on the plane R 2 are non-euclidean. Let s denote arc length. Then Euclidean geometry arises from the

### Students will understand 1. use numerical bases and the laws of exponents

Grade 8 Expressions and Equations Essential Questions: 1. How do you use patterns to understand mathematics and model situations? 2. What is algebra? 3. How are the horizontal and vertical axes related?

### 2.1 Three Dimensional Curves and Surfaces

. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The

### FURTHER VECTORS (MEI)

Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

### Mathematics 1. Lecture 5. Pattarawit Polpinit

Mathematics 1 Lecture 5 Pattarawit Polpinit Lecture Objective At the end of the lesson, the student is expected to be able to: familiarize with the use of Cartesian Coordinate System. determine the distance

### TWO-DIMENSIONAL TRANSFORMATION

CHAPTER 2 TWO-DIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization

### * Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B. PPT No.

* Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B PPT No. 17 Biot Savart s Law A straight infinitely long wire is carrying

### 6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)

6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,

### 3. Double Integrals 3A. Double Integrals in Rectangular Coordinates

3. Double Integrals 3A. Double Integrals in ectangular Coordinates 3A-1 Evaluate each of the following iterated integrals: c) 2 1 1 1 x 2 (6x 2 +2y)dydx b) x 2x 2 ydydx d) π/2 π 1 u (usint+tcosu)dtdu u2

### 0017- Understanding and Using Vector and Transformational Geometries

Tom Coleman INTD 301 Final Project Dr. Johannes Vector geometry: 0017- Understanding and Using Vector and Transformational Geometries 3-D Cartesian coordinate representation: - A vector v is written as

### Higher Geometry Problems

Higher Geometry Problems ( Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement

### Sphere centered at the origin.

A Quadratic surfaces In this appendix we will study several families of so-called quadratic surfaces, namely surfaces z = f(x, y) which are defined by equations of the type A + By 2 + Cz 2 + Dxy + Exz

### ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

### Equations Involving Lines and Planes Standard equations for lines in space

Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity

Section.1 Radian Measure Another way of measuring angles is with radians. This allows us to write the trigonometric functions as functions of a real number, not just degrees. A central angle is an angle

### 4.1 Radian and Degree Measure

Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position