where C is traversed in the clockwise direction, r 5 èuè =h, sin u; cos ui; u ë0; çè; è6è where C is traversed in the counterclockwise direction èhow
|
|
|
- Arlene Briggs
- 6 years ago
- Views:
Transcription
1 1 A Note on Parametrization The key to parametrizartion is to realize that the goal of this method is to describe the location of all points on a geometric object, a curve, a surface, or a region. This description must be one-to-one and onto: every point must be described once and only once. 1 Parametrization of Curves in R Let us begin with parametrizing the curve C whose equation is given by x + y =4 è1è i.e., a circle of radius centered at the origin. We start by associating a position vector r to each point èx; yè onc through the relation r = hx; yi: èè The coordinates x and y in èèare not arbitrary í they are related through equation è1è. This means that we are free to assign a value to only one of the coordinates of a typical point on C; the other coordinate must be determined from the equation of the circle. For this reason we say C has one degree of freedom. Choosing x as the parameter for C, we see from è1è that y = æp 4, x ; where the positive square root describes those points on C that lie above the x-axis and the negative square root the points below the x-axis. The complete parametrization of C is r 1 èxè =hx;p 4, x i and r èxè =hx;,p 4, x i; è3è where, ç x ç for r 1 and, x for r. Note that the points è,; 0è and è; 0è are arbitrarily assigned to r 1. We can now use the parametrization of C to determine tangent vectors to C, plot C on a graphics software, or to perform a line integral around C. Although the paramerization in è3è is adequate for the purpose of describing C, it is not the most convenient description of this curve. A more eæcient way to view C is to use polar coordinates to describe its points: x = cos ç; y = sin ç, with ç ë0; çè. So C can also be parametrized as r 3 èçè =h cos ç; sin çi; ç ë0; çè: è4è Note that r 3 in è4è does the job of both r 1 and r in è3è. The parametrizations r 1, r and r 3 are just a few ways out of the inænitely many ways that one could describe C. Here are three other parametrizations of the same curve: r 4 ètè =h sin t; cos ti; t ë0; çè; è5è
2 where C is traversed in the clockwise direction, r 5 èuè =h, sin u; cos ui; u ë0; çè; è6è where C is traversed in the counterclockwise direction èhow is r 5 diæerent from r 3?è and r 6 èwè =h sin w; cos wi; w ë0;çè: è7è To understand the diæerence between r 4 and r 6, compute the speed of a particle traveling around C according to these parametrizations. Let us now consider parametrizations of other familiar curves. Any two dimensional curve whose equation is given by y = fèxè can be parametrized as rèxè =hx; fèxèi; x èa; bè; èè so, for instance, the straight line y = mx + b can be viewed as rèxè = hx; mx + bi: è9è The circle of radius a centered at èb; cè is parametrized as rèçè =hb + a cos ç; c + a sin çi; ç è0; çë: è10è The ellipse whose equation is given by a x + b y = c is parametrized as èto see where the following expressions come from, divide a x + b y = c by c and set the term containing x equal to cos t and the one containing y to sin tè rètè =h c a cos t; c sin ti t è0; çè: è11è b Parametrization of Curves in R 3 Similar to curves in R, curves in R 3 still have only one degree of freedom, that is, a single parameter is suæcient to describe the coordinates of a typical point on curves in R 3. As an example, consider the straight line C that connects the two points P =è1; ; 1è and Q =è,1; 1; 3è. Let P = h1; ; 1i and Q = h,1; 1; 3i. Deæne v = Q, P = h,;,1; i. Note that v is parallel to the line C. So every point S on C can be accessed by the vector S = P + tv for some t R. So rètè =h1; ; 1i + th,;,1; i; t R è1è is a parametrization of C. In terms of coordinates, è1è is equivalent to : xètè = 1, t yètè =, t zètè = 1+t è13è
3 3 Every straight line C, whether in R or R 3, can be parametrized as rètè =r 0 + tv; t R è14è where r 0 is the position vector corresponding to a known point onc èsuch as h1; ; 1i in our previous exampleè, and v is a vector parallel to C. For instance, to ænd the parametrization of the line of intersection between the two planes x, 3y + z = and x + y + z = 0, ærst we ænd a point on this line by setting z = 0 in the equations of the planes and then solve for x and y to see that è 5 ;, ; 0è lies on C. Next, we note that 5 the vectors n 1 = h;,3; 1i and n = h1; 1; 1i are normal to the planes. Therefore, is parallel to C. Therefore v = n 1 æ n = h,4;,1; 5i rètè =h 5 ;, ; 0i + th,4;,1; 5i 5 è15è is a parametrization of C. More complicated curves are parametrized similarly. Typical points on a curve C are accessed by a position vector r of the form rètè =hxètè;yètè;zètèi: For example, the parametrization hsin t; cos t; ti describes a helix in R 3. Or the intersection of the plane x + y + z = 1 and the cylinder x + y = 1 is given by rètè =hcos t; sin t; 1, cos t, sin ti; t è0; çë: è16è 3 Parametrization of Surfaces Surfaces in R 3 are characterized by two degrees of freedom; one is allowed to vary two parameters independently to cover all points on a surface. The simplest examples are surfaces that are graphs of functions f that depend on two variables, z = fèx; yè. Such surfaces are often parametrized as rèx; yè =hx; y; fèx; yèi; axb; cyd: è17è For example, the surface z = x + y over the unit square is parametrized as rèx; yè =hx; y; x + y i; 0 x1; 0 y1: è1è The cylinder x + y = 1 is parametrized as rèç; zè =hcos ç; sin ç; zi; ç è0; çë; z R; è19è while the cylinder x + z = 4 is parametrized as rèç; yè =h cos ç; y; sin çi; ç è0; çë; y R; è0è
4 4 The surface of the disk of radius a in the plane z = b centered at the origin is given by rèu; vè =hu cos v; u sin v; bi; u ë0; 1ë; v è0; çë: è1è Certain surfaces are best parametrized in spherical coordinates where : x = ç cos ç sin ç; y = ç sin ç sin ç; z = ç cos ç: èè For example, the cone z = x + y can be parametrized as rèç; çè = p hç cos ç; ç sin ç; çi; ç R; ç è0; çë: è3è Similarly, the northern hemisphere of radius 3 centered at the origin may be parametrized as rèç; çè =3hcos ç sin ç; sin ç sin ç; cos çi; ç è0; çë; ç ë0; ç ë: è4è An alternative way of parametrizing this surface is as follows: q rèx; yè =3hx; y; 9, x, y i; x + y ç 9: è5è The boundary of this surface èthe circle of radius 3 in the xy-plane and centered at the originè is best parametrized using è4è by setting ç = ç in that relation to get rèçè = 3hcos ç; sin ç; 0i; ç è0; çë: è6è Once a parametrization rèu; vè of a surface S is known, the vector deænes a normal vector to S. r u æ r v 4 Parametrization of Regions in R 3 Regions in R 3 have three degrees of freedom. They are parametrized by rèu; v; wè where u, v and w take onvalues in respective intervals. For example, the region bounded by the cylinder x + y = 1 and the planes z =, and z = 1 is paramterized as rèr;ç;zè=hr cos ç; r sin ç; zi; 0 ç r ç 1; 0 ç ç ç ç;, ç z ç 1: è7è The boundary of this region consists of three surfaces S 1, S and S 3 given by : S 1 : r 1 èu; vè = hu cos v; u sin v;,i; 0 ç u ç 1; 0 ç vç; S : r èu; vè = hu cos v; u sin v; 1i; 0 ç u ç 1; 0 ç vç; S 3 : r 3 èu; vè = hcos v; sin v; ui;, ç u ç 1; 0 ç v:ç: èè
5 5 Similarly, the region inside the northern hemisphere of radius is parametrized as follows: rèç; ç; çè =hç cos ç sin ç; ç sin ç sin ç; ç cos çi; 0 ç ç ç ; 0 ç çç; 0 ç ç ç ç è9è The boundary of this region consists of two surfaces S 1 and S given by è S 1 : r 1 èç; çè = hcos ç sin ç; sin ç sin ç; cos çi; 0 ç çç; 0 ç ç ç ; S : r èr;çè = hr cos ç; r sin ç; 0i; 0 ç r ç ; 0 ç çç: è30è
Math 241: Calculus III Final Exam è300 points totalè Show All Work Name This test consists of 2 parts. There are 10 problems in Part I, each worth 18
Math 241: Calculus III Final Exam è300 points totalè Show All Work Name This test consists of 2 parts. There are 10 problems in Part I, each worth 18 points; and there are 5 problems in Part II, each worth
Curvilinear Coordinates
Curvilinear Coordinates Cylindrical Coordinates A 3-dimensional coordinate transformation is a mapping of the form T (u; v; w) = hx (u; v; w) ; y (u; v; w) ; z (u; v; w)i Correspondingly, a 3-dimensional
Name: Final Exam Review. (b) Reparameterize r(t) with respect to arc length measured for the point (1, 0, 1) in the direction of increasing t.
MATH 127 ALULU III Name: 1. Let r(t) = e t i + e t sin t j + e t cos t k (a) Find r (t) Final Exam Review (b) Reparameterize r(t) with respect to arc length measured for the point (1,, 1) in the direction
16.6 Parametric Surfaces and Their Areas
SECTION 6.6 PARAMETRIC SURFACES AND THEIR AREAS i j k (b) From (a), v = w r = =( ) i +( ) j +( ) k = i + j i j k (c) curl v = v = = () () i + ( ) () j + () ( ) k =[ ( )] k = k =w 9. For any continuous
MATH 234. Excercises on Integration in Several Variables. I. Double Integrals
MATH 234 Excercises on Integration in everal Variables I. Double Integrals Problem 1. D = {(x, y) : y x 1, 0 y 1}. Compute D ex3 da. Problem 2. Find the volume of the solid bounded above by the plane 3x
Chapter 11. Parametric Equations And Polar Coordinates
Instructor: Prof. Dr. Ayman H. Sakka Chapter 11 Parametric Equations And Polar Coordinates In this chapter we study new ways to define curves in the plane, give geometric definitions of parabolas, ellipses,
MATH 1020 WORKSHEET 10.1 Parametric Equations
MATH WORKSHEET. Parametric Equations If f and g are continuous functions on an interval I, then the equations x ft) and y gt) are called parametric equations. The parametric equations along with the graph
PART I. Answer each of the following. è1è Let! u = h2;,1;,2i and! v = h3; 1;,1i. Calculate: èaè! u, 2! v èbè the dot product of! u and! v ècè! u æ! v
MATH 241: FINAL EXAM Name Instructions and Point Values: Put your name in the space provided above. Check that your test contains 14 diæerent pages including one blank page. Work each problem below and
Lecture 34: Curves defined by Parametric equations
Curves defined by Parametric equations When the path of a particle moving in the plane is not the graph of a function, we cannot describe it using a formula that express y directly in terms of x, or x
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = enter NONE.
Ch17 Practice Test Sketch the vector field F. F(x, y) = (x - y)i + xj Evaluate the line integral, where C is the given curve. C xy 4 ds. C is the right half of the circle x 2 + y 2 = 4 oriented counterclockwise.
Parametric Surfaces. Substitution
Calculus Lia Vas Parametric Surfaces. Substitution Recall that a curve in space is given by parametric equations as a function of single parameter t x = x(t) y = y(t) z = z(t). A curve is a one-dimensional
16.6. Parametric Surfaces. Parametric Surfaces. Parametric Surfaces. Vector Calculus. Parametric Surfaces and Their Areas
16 Vector Calculus 16.6 and Their Areas Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. and Their Areas Here we use vector functions to describe more general
Name: Class: Date: 1. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint.
. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. f (x, y) = x y, x + y = 8. Set up the triple integral of an arbitrary continuous function
Surfaces. U (x; y; z) = k. Indeed, many of the most familiar surfaces are level surfaces of functions of 3 variables.
Surfaces Level Surfaces One of the goals of this chapter is to use di erential calculus to explore surfaces, in much the same way that we used di erential calculus to study curves in the rst chapter. In
To graph the point (r, θ), simply go out r units along the initial ray, then rotate through the angle θ. The point (1, 5π 6. ) is graphed below:
Polar Coordinates Any point in the plane can be described by the Cartesian coordinates (x, y), where x and y are measured along the corresponding axes. However, this is not the only way to represent points
f xx (x, y) = 6 + 6x f xy (x, y) = 0 f yy (x, y) = y In general, the quantity that we re interested in is
1. Let f(x, y) = 5 + 3x 2 + 3y 2 + 2y 3 + x 3. (a) Final all critical points of f. (b) Use the second derivatives test to classify the critical points you found in (a) as a local maximum, local minimum,
PARAMETERIZATIONS OF PLANE CURVES
PARAMETERIZATIONS OF PLANE CURVES Suppose we want to plot the path of a particle moving in a plane. This path looks like a curve, but we cannot plot it like we would plot any other type of curve in the
Math 52 Final Exam March 16, 2009
Math 52 Final Exam March 16, 2009 Name : Section Leader: Josh Lan Xiannan (Circle one) Genauer Huang Li Section Time: 10:00 11:00 1:15 2:15 (Circle one) This is a closed-book, closed-notes exam. No calculators
Jim Lambers MAT 169 Fall Semester Lecture 33 Notes
Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 33 Notes These notes correspond to Section 9.3 in the text. Polar Coordinates Throughout this course, we have denoted a point in the plane by an ordered
The Three Dimensional Coordinate System
The Three-Dimensional Coordinate System The Three Dimensional Coordinate System You can construct a three-dimensional coordinate system by passing a z-axis perpendicular to both the x- and y-axes at the
Parametric Surfaces and Surface Area
Parametric Surfaces and Surface Area What to know: 1. Be able to parametrize standard surfaces, like the ones in the handout.. Be able to understand what a parametrized surface looks like (for this class,
UNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL 5
UNIVERSITI TEKNOLOGI MALAYSIA SSE 189 ENGINEERING MATHEMATIS TUTORIAL 5 1. Evaluate the following surface integrals (i) (x + y) ds, : part of the surface 2x+y+z = 6 in the first octant. (ii) (iii) (iv)
6.7. POLAR COORDINATES
6.7. POLAR COORDINATES What You Should Learn Plot points on the polar coordinate system. Convert points from rectangular to polar form and vice versa. Convert equations from rectangular to polar form and
PARAMETRIC EQUATIONS AND POLAR COORDINATES
10 PARAMETRIC EQUATIONS AND POLAR COORDINATES PARAMETRIC EQUATIONS & POLAR COORDINATES A coordinate system represents a point in the plane by an ordered pair of numbers called coordinates. PARAMETRIC EQUATIONS
MATH 200 (Fall 2016) Exam 1 Solutions (a) (10 points) Find an equation of the sphere with center ( 2, 1, 4).
MATH 00 (Fall 016) Exam 1 Solutions 1 1. (a) (10 points) Find an equation of the sphere with center (, 1, 4). (x ( )) + (y 1) + (z ( 4)) 3 (x + ) + (y 1) + (z + 4) 9 (b) (10 points) Find an equation of
Multivariable Calculus
Multivariable Calculus Chapter 10 Topics in Analytic Geometry (Optional) 1. Inclination of a line p. 5. Circles p. 4 9. Determining Conic Type p. 13. Angle between lines p. 6. Parabolas p. 5 10. Rotation
2. Give an example of a non-constant function f(x, y) such that the average value of f over is 0.
Midterm 3 Review Short Answer 2. Give an example of a non-constant function f(x, y) such that the average value of f over is 0. 3. Compute the Riemann sum for the double integral where for the given grid
You may know these...
You may know these... Chapter 1: Multivariables Functions 1.1 Functions of Two Variables 1.1.1 Function representations 1.1. 3-D Coordinate System 1.1.3 Graph of two variable functions 1.1.4 Sketching
Getting a New Perspective
Section 6.3 Polar Coordinates Getting a New Perspective We have worked etensively in the Cartesian coordinate system, plotting points, graphing equations, and using the properties of the Cartesian plane
Complex Numbers, Polar Equations, and Parametric Equations. Copyright 2017, 2013, 2009 Pearson Education, Inc.
8 Complex Numbers, Polar Equations, and Parametric Equations Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 8.5 Polar Equations and Graphs Polar Coordinate System Graphs of Polar Equations Conversion
Grad operator, triple and line integrals. Notice: this material must not be used as a substitute for attending the lectures
Grad operator, triple and line integrals Notice: this material must not be used as a substitute for attending the lectures 1 .1 The grad operator Let f(x 1, x,..., x n ) be a function of the n variables
Calculus III. Math 233 Spring In-term exam April 11th. Suggested solutions
Calculus III Math Spring 7 In-term exam April th. Suggested solutions This exam contains sixteen problems numbered through 6. Problems 5 are multiple choice problems, which each count 5% of your total
9.5 Polar Coordinates. Copyright Cengage Learning. All rights reserved.
9.5 Polar Coordinates Copyright Cengage Learning. All rights reserved. Introduction Representation of graphs of equations as collections of points (x, y), where x and y represent the directed distances
Find the specific function values. Complete parts (a) through (d) below. f (x,y,z) = x y y 2 + z = (Simplify your answer.) ID: 14.1.
. Find the specific function values. Complete parts (a) through (d) below. f (x,y,z) = x y y 2 + z 2 (a) f(2, 4,5) = (b) f 2,, 3 9 = (c) f 0,,0 2 (d) f(4,4,00) = = ID: 4..3 2. Given the function f(x,y)
14.5 Directional Derivatives and the Gradient Vector
14.5 Directional Derivatives and the Gradient Vector 1. Directional Derivatives. Recall z = f (x, y) and the partial derivatives f x and f y are defined as f (x 0 + h, y 0 ) f (x 0, y 0 ) f x (x 0, y 0
Double Integrals, Iterated Integrals, Cross-sections
Chapter 14 Multiple Integrals 1 ouble Integrals, Iterated Integrals, Cross-sections 2 ouble Integrals over more general regions, efinition, Evaluation of ouble Integrals, Properties of ouble Integrals
6. Find the equation of the plane that passes through the point (-1,2,1) and contains the line x = y = z.
Week 1 Worksheet Sections from Thomas 13 th edition: 12.4, 12.5, 12.6, 13.1 1. A plane is a set of points that satisfies an equation of the form c 1 x + c 2 y + c 3 z = c 4. (a) Find any three distinct
Chapter 10 Homework: Parametric Equations and Polar Coordinates
Chapter 1 Homework: Parametric Equations and Polar Coordinates Name Homework 1.2 1. Consider the parametric equations x = t and y = 3 t. a. Construct a table of values for t =, 1, 2, 3, and 4 b. Plot the
the straight line in the xy plane from the point (0, 4) to the point (2,0)
Math 238 Review Problems for Final Exam For problems #1 to #6, we define the following paths vector fields: b(t) = the straight line in the xy plane from the point (0, 4) to the point (2,0) c(t) = the
ENGI Parametric & Polar Curves Page 2-01
ENGI 3425 2. Parametric & Polar Curves Page 2-01 2. Parametric and Polar Curves Contents: 2.1 Parametric Vector Functions 2.2 Parametric Curve Sketching 2.3 Polar Coordinates r f 2.4 Polar Curve Sketching
A1:Orthogonal Coordinate Systems
A1:Orthogonal Coordinate Systems A1.1 General Change of Variables Suppose that we express x and y as a function of two other variables u and by the equations We say that these equations are defining a
MAC2313 Final A. a. The vector r u r v lies in the tangent plane of S at a given point. b. S f(x, y, z) ds = R f(r(u, v)) r u r v du dv.
MAC2313 Final A (5 pts) 1. Let f(x, y, z) be a function continuous in R 3 and let S be a surface parameterized by r(u, v) with the domain of the parameterization given by R; how many of the following are
Applications of Triple Integrals
Chapter 14 Multiple Integrals 1 Double Integrals, Iterated Integrals, Cross-sections 2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals
To graph the point (r, θ), simply go out r units along the initial ray, then rotate through the angle θ. The point (1, 5π 6
Polar Coordinates Any point in the plane can be described by the Cartesian coordinates (x, y), where x and y are measured along the corresponding axes. However, this is not the only way to represent points
Topic 5-6: Parameterizing Surfaces and the Surface Elements ds and ds. Big Ideas. What We Are Doing Today... Notes. Notes. Notes
Topic 5-6: Parameterizing Surfaces and the Surface Elements ds and ds. Textbook: Section 16.6 Big Ideas A surface in R 3 is a 2-dimensional object in 3-space. Surfaces can be described using two variables.
Math 126C: Week 3 Review
Math 126C: Week 3 Review Note: These are in no way meant to be comprehensive reviews; they re meant to highlight the main topics and formulas for the week. Doing homework and extra problems is always the
Chapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
MATH 116 REVIEW PROBLEMS for the FINAL EXAM
MATH 116 REVIEW PROBLEMS for the FINAL EXAM The following questions are taken from old final exams of various calculus courses taught in Bilkent University 1. onsider the line integral (2xy 2 z + y)dx
Polar (BC Only) They are necessary to find the derivative of a polar curve in x- and y-coordinates. The derivative
Polar (BC Only) Polar coordinates are another way of expressing points in a plane. Instead of being centered at an origin and moving horizontally or vertically, polar coordinates are centered at the pole
1) Find. a) b) c) d) e) 2) The function g is defined by the formula. Find the slope of the tangent line at x = 1. a) b) c) e) 3) Find.
1 of 7 1) Find 2) The function g is defined by the formula Find the slope of the tangent line at x = 1. 3) Find 5 1 The limit does not exist. 4) The given function f has a removable discontinuity at x
Coordinate Transformations in Advanced Calculus
Coordinate Transformations in Advanced Calculus by Sacha Nandlall T.A. for MATH 264, McGill University Email: sacha.nandlall@mail.mcgill.ca Website: http://www.resanova.com/teaching/calculus/ Fall 2006,
5. y 2 + z 2 + 4z = 0 correct. 6. z 2 + x 2 + 2x = a b = 4 π
M408D (54690/95/00), Midterm #2 Solutions Multiple choice questions (20 points) See last two pages. Question #1 (25 points) Dene the vector-valued function r(t) = he t ; 2; 3e t i: a) At what point P (x
Math 265 Exam 3 Solutions
C Roettger, Fall 16 Math 265 Exam 3 Solutions Problem 1 Let D be the region inside the circle r 5 sin θ but outside the cardioid r 2 + sin θ. Find the area of D. Note that r and θ denote polar coordinates.
Test Name: Chapter 3 Review
Test Name: Chapter 3 Review 1. For the following equation, determine the values of the missing entries. If needed, write your answer as a fraction reduced to lowest terms. 10x - 8y = 18 Note: Each column
Mathematically, the path or the trajectory of a particle moving in space in described by a function of time.
Module 15 : Vector fields, Gradient, Divergence and Curl Lecture 45 : Curves in space [Section 45.1] Objectives In this section you will learn the following : Concept of curve in space. Parametrization
Advanced Algebra. Equation of a Circle
Advanced Algebra Equation of a Circle Task on Entry Plotting Equations Using the table and axis below, plot the graph for - x 2 + y 2 = 25 x -5-4 -3 0 3 4 5 y 1 4 y 2-4 3 2 + y 2 = 25 9 + y 2 = 25 y 2
1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral
1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral Show your working and give your answer correct to three decimal places. 2 2.5 3 3.5 4 When When When When When
Section 10.1 Polar Coordinates
Section 10.1 Polar Coordinates Up until now, we have always graphed using the rectangular coordinate system (also called the Cartesian coordinate system). In this section we will learn about another system,
= f (a, b) + (hf x + kf y ) (a,b) +
Chapter 14 Multiple Integrals 1 Double Integrals, Iterated Integrals, Cross-sections 2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals
Vectors and the Geometry of Space
Vectors and the Geometry of Space In Figure 11.43, consider the line L through the point P(x 1, y 1, z 1 ) and parallel to the vector. The vector v is a direction vector for the line L, and a, b, and c
HOUGH TRANSFORM CS 6350 C V
HOUGH TRANSFORM CS 6350 C V HOUGH TRANSFORM The problem: Given a set of points in 2-D, find if a sub-set of these points, fall on a LINE. Hough Transform One powerful global method for detecting edges
MATH 2023 Multivariable Calculus
MATH 2023 Multivariable Calculus Problem Sets Note: Problems with asterisks represent supplementary informations. You may want to read their solutions if you like, but you don t need to work on them. Set
How to Project Spherical Conics into the Plane
How to Project pherical Conics into the Plane Yoichi Maeda maeda@keyakiccu-tokaiacjp Department of Mathematics Tokai University Japan Abstract: In this paper, we will introduce a method how to draw the
MATH 2400: CALCULUS 3 MAY 9, 2007 FINAL EXAM
MATH 4: CALCULUS 3 MAY 9, 7 FINAL EXAM I have neither given nor received aid on this exam. Name: 1 E. Kim................ (9am) E. Angel.............(1am) 3 I. Mishev............ (11am) 4 M. Daniel...........
7-5 Parametric Equations
3. Sketch the curve given by each pair of parametric equations over the given interval. Make a table of values for 6 t 6. t x y 6 19 28 5 16.5 17 4 14 8 3 11.5 1 2 9 4 1 6.5 7 0 4 8 1 1.5 7 2 1 4 3 3.5
Lagrange multipliers October 2013
Lagrange multipliers 14.8 14 October 2013 Example: Optimization with constraint. Example: Find the extreme values of f (x, y) = x + 2y on the ellipse 3x 2 + 4y 2 = 3. 3/2 1 1 3/2 Example: Optimization
Three-Dimensional Coordinate Systems
Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 17 Notes These notes correspond to Section 10.1 in the text. Three-Dimensional Coordinate Systems Over the course of the next several lectures, we will
4 = 1 which is an ellipse of major axis 2 and minor axis 2. Try the plane z = y2
12.6 Quadrics and Cylinder Surfaces: Example: What is y = x? More correctly what is {(x,y,z) R 3 : y = x}? It s a plane. What about y =? Its a cylinder surface. What about y z = Again a cylinder surface
WHAT YOU SHOULD LEARN
GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of
Chapter 6 Some Applications of the Integral
Chapter 6 Some Applications of the Integral More on Area More on Area Integrating the vertical separation gives Riemann Sums of the form More on Area Example Find the area A of the set shaded in Figure
Each point P in the xy-plane corresponds to an ordered pair (x, y) of real numbers called the coordinates of P.
Lecture 7, Part I: Section 1.1 Rectangular Coordinates Rectangular or Cartesian coordinate system Pythagorean theorem Distance formula Midpoint formula Lecture 7, Part II: Section 1.2 Graph of Equations
Rectangular Coordinates in Space
Rectangular Coordinates in Space Philippe B. Laval KSU Today Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 1 / 11 Introduction We quickly review one and two-dimensional spaces and then
Math 253, Section 102, Fall 2006 Practice Final Solutions
Math 253, Section 102, Fall 2006 Practice Final Solutions 1 2 1. Determine whether the two lines L 1 and L 2 described below intersect. If yes, find the point of intersection. If not, say whether they
We can set up the integral over this elliptical region as a y-simple region: This integral can be evaluated as follows. The inner integral is
Volume using Double Integrals Eample : Find the volume of the region in R that is under the paraboloid z = + y, above the y plane and below z =. Solution: The region is shown below. The planes z = and
SPECIAL TECHNIQUES-II
SPECIAL TECHNIQUES-II Lecture 19: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Method of Images for a spherical conductor Example :A dipole near aconducting sphere The
Background for Surface Integration
Background for urface Integration 1 urface Integrals We have seen in previous work how to define and compute line integrals in R 2. You should remember the basic surface integrals that we will need to
Lagrange multipliers 14.8
Lagrange multipliers 14.8 14 October 2013 Example: Optimization with constraint. Example: Find the extreme values of f (x, y) = x + 2y on the ellipse 3x 2 + 4y 2 = 3. 3/2 Maximum? 1 1 Minimum? 3/2 Idea:
Mysterious or unsupported answers will not receive full credit. Your work should be mathematically correct and carefully and legibly written.
Math 2374 Spring 2006 Final May 8, 2006 Time Limit: 1 Hour Name (Print): Student ID: Section Number: Teaching Assistant: Signature: This exams contains 11 pages (including this cover page) and 10 problems.
13.1. Functions of Several Variables. Introduction to Functions of Several Variables. Functions of Several Variables. Objectives. Example 1 Solution
13 Functions of Several Variables 13.1 Introduction to Functions of Several Variables Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives Understand
Parametric and Polar Curves
Chapter 2 Parametric and Polar Curves 2.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves Parametric Equations So far we ve described a curve by giving an equation that the coordinates
Parametric and Polar Curves
Chapter 2 Parametric and Polar Curves 2.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves Parametric Equations So far we ve described a curve by giving an equation that the coordinates
18.02 Final Exam. y = 0
No books, notes or calculators. 5 problems, 50 points. 8.0 Final Exam Useful formula: cos (θ) = ( + cos(θ)) Problem. (0 points) a) (5 pts.) Find the equation in the form Ax + By + z = D of the plane P
Quadric Surfaces. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) Quadric Surfaces Spring /
.... Quadric Surfaces Philippe B. Laval KSU Spring 2012 Philippe B. Laval (KSU) Quadric Surfaces Spring 2012 1 / 15 Introduction A quadric surface is the graph of a second degree equation in three variables.
Solved Examples. Parabola with vertex as origin and symmetrical about x-axis. We will find the area above the x-axis and double the area.
Solved Examples Example 1: Find the area common to the curves x 2 + y 2 = 4x and y 2 = x. x 2 + y 2 = 4x (i) (x 2) 2 + y 2 = 4 This is a circle with centre at (2, 0) and radius 2. y = (4x-x 2 ) y 2 = x
Graphics and Interaction Transformation geometry and homogeneous coordinates
433-324 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
Quadric Surfaces. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Quadric Surfaces Today 1 / 24
Quadric Surfaces Philippe B. Laval KSU Today Philippe B. Laval (KSU) Quadric Surfaces Today 1 / 24 Introduction A quadric surface is the graph of a second degree equation in three variables. The general
53 M 0 j èm 2 i è ;1 M 0 j èm 2 i è ;1 èm 2 i è ;0 èm 2 i è ;0 (a) (b) M 0 i M 0 i (c) (d) Figure 6.1: Invalid boundary layer elements due to invisibi
CHAPTER 6 BOUNDARY LAYER MESHING - ENSURING ELEMENT VALIDITY Growth curves are created with maximum consideration for topological validity and topological compatibility with the model. However, only preliminary
COMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates
COMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
Polar Coordinates. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 27 / 45
: Given any point P = (x, y) on the plane r stands for the distance from the origin (0, 0). θ stands for the angle from positive x-axis to OP. Polar coordinate: (r, θ) Chapter 10: Parametric Equations
Math 2374 Spring 2007 Midterm 3 Solutions - Page 1 of 6 April 25, 2007
Math 374 Spring 7 Midterm 3 Solutions - Page of 6 April 5, 7. (3 points) Consider the surface parametrized by (x, y, z) Φ(x, y) (x, y,4 (x +y )) between the planes z and z 3. (i) (5 points) Set up the
Volumes of Solids of Revolution Lecture #6 a
Volumes of Solids of Revolution Lecture #6 a Sphereoid Parabaloid Hyperboloid Whateveroid Volumes Calculating 3-D Space an Object Occupies Take a cross-sectional slice. Compute the area of the slice. Multiply
1. Suppose that the equation F (x, y, z) = 0 implicitly defines each of the three variables x, y, and z as functions of the other two:
Final Solutions. Suppose that the equation F (x, y, z) implicitly defines each of the three variables x, y, and z as functions of the other two: z f(x, y), y g(x, z), x h(y, z). If F is differentiable
Surfaces and Integral Curves
MODULE 1: MATHEMATICAL PRELIMINARIES 16 Lecture 3 Surfaces and Integral Curves In Lecture 3, we recall some geometrical concepts that are essential for understanding the nature of solutions of partial
f sin the slope of the tangent line is given by f sin f cos cos sin , but it s also given by 2. So solve the DE with initial condition: sin cos
Math 414 Activity 1 (Due by end of class August 1) 1 Four bugs are placed at the four corners of a square with side length a The bugs crawl counterclockwise at the same speed and each bug crawls directly
MA 243 Calculus III Fall Assignment 1. Reading assignments are found in James Stewart s Calculus (Early Transcendentals)
MA 43 Calculus III Fall 8 Dr. E. Jacobs Assignments Reading assignments are found in James Stewart s Calculus (Early Transcendentals) Assignment. Spheres and Other Surfaces Read. -. and.6 Section./Problems
Direction Fields; Euler s Method
Direction Fields; Euler s Method It frequently happens that we cannot solve first order systems dy (, ) dx = f xy or corresponding initial value problems in terms of formulas. Remarkably, however, this
, etc. Let s work with the last one. We can graph a few points determined by this equation.
1. Lines By a line, we simply mean a straight curve. We will always think of lines relative to the cartesian plane. Consider the equation 2x 3y 4 = 0. We can rewrite it in many different ways : 2x 3y =
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Calculus III-Final review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the corresponding position vector. 1) Define the points P = (-,
Worksheet 3.4: Triple Integrals in Cylindrical Coordinates. Warm-Up: Cylindrical Volume Element d V
Boise State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates From the Toolbox (what you need from previous classes) Know what the volume element dv represents. Be able to find
Unit 12 Topics in Analytic Geometry - Classwork
Unit 1 Topics in Analytic Geometry - Classwork Back in Unit 7, we delved into the algebra and geometry of lines. We showed that lines can be written in several forms: a) the general form: Ax + By + C =