MTH234 Chapter 16 - Vector Calculus Michigan State University
|
|
- Horace Bennett
- 5 years ago
- Views:
Transcription
1 MTH234 hapter 6 - Vector alculus Michigan State Universit 4 Green s Theorem Green s Theorem gives a relationship between double integrals and line integrals around simple closed curves. (Start and end at the same point. Are not self-intersecting ecept at endpoints.) Draw picture of region D with boundar. Positive and negative orientation. Definition(s) 4... A simple closed curve has positive orientation if its parametrization traverses the curve eactl once in a counterclockwise direction. 2. A simple closed curve has negative orientation if its parametrization traverses the curve eactl once in a clockwise direction. Theorem 4.2. Let be a positivel oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded b. If F P, Q have continuous partial derivatives on an open region that contains D then, P d + Q d D ( Q P ) da or equivalentl F T ds D ( Q P ) da The idea of the proof is important because it will come up again in Stokes Theorem. The idea is circulation. Because we have a closed simple curve the integral F T ds counts how the particles on the curve are circulating. Green s Theorem sas that instead of counting how picture here. breaking up and canceling ecept at edges the particles are circulating on the curve we can count how the particles are circulating inside the curve.
2 MTH234 hapter 6 - Vector alculus Michigan State Universit That is (irculation of points on curve) (irculation of points inside curve) Idea of Proof So now we need to determine circulation at a point. consider circulation around small rectangles. Along the 4 boundaries of the rectangle we get: First lets Top: F(, + ) ( i) P (, + ) (, + ) F ( i) ( +, + ) Bottom: F(, ) (i) P (, ) F ( j) F (j) Right: F( +, ) j Q( +, ) (, ) F (i) ( +, ) Left: F(, ) ( j) Q(, ) Grouping favorabl we get: irculation of Top + Bottom + Right + Left irculation of P (, + ) + P (, ) + Q( +, ) + Q(, ) irculation of ( P (, + ) + P (, )) + Q( +, ) Q(, ) irculation of ( P + Q ) Now we need to scale from circulation on a rectangle to circulation at a point irculation at irculation of Area of irculation at ( P + Q ) Q P Q P And so now we are read to see wh we love Green s Theorem 2
3 MTH234 hapter 6 - Vector alculus Michigan State Universit Eample 4.3. Find the work done b F 4 2, 2 4 once counterclockwise around the curve given b the picture: Solution. Let s pretend we forgot Green s Theorem on the eam. 3 To parametrize this curve correctl I need to break it into 4 pieces F T ds B F T ds + F T ds + F T ds + F T ds T LL RL Parametrizing the four pieces we see that (in a counterclockwise direction) B : r(t) cos t, sin t t [π, 0] r (t) sin t, cos t T : r(t) 3 cos t, 3 sin t t [0, π] r (t) sin t, 3 cos t LL : r(t) t, 0 t [, 3] r (t), 0 RL : r(t) t, 0 t [, ] r (t), 0 Let s calculate these individual integrals 0 B F T ds π 4 2, 2 4 r (t) dt 0 4(cos t) 2(sin t), 2(cos t) 4(sin t) sin t, cos t dt π 0 π 4 sin t cos t + 2 sin2 t + 2 cos 2 t 4 sin t cos t dt 0 8 sin t cos t + 2 dt π 2(0 π) 2π π T F T ds 0 4 2, 2 4 r (t) dt π 4(3 cos t) 2(3 sin t), 2(3 cos t) 4(3 sin t) sin t, 3 cos t dt 0 π 0 6 sin t cos t + 8 sin2 t + 8 cos 2 t 36 sin t cos t dt π 72 sin t cos t + 8 dt 0 8(π 0) 8π LL F T ds 4 2, 2 4 r (t) dt 4(t) 2(0), 2(t) 4(0), 0 dt 4t dt [ 2t 2] 2( 9) 6 3 RL F T ds 4 2, 2 4 r (t) dt 3 4(t) 2(0), 2(t) 4(0), 0 dt 3 4t dt [ 2t 2] 3 2(9 ) 6 Giving us our final answer of F T ds 2π + 8π π Now let s imagine ou remember Green s Theorem. 3
4 MTH234 hapter 6 - Vector alculus Michigan State Universit Eample 4.3. Find the work done b F 4 2, 2 4 once counterclockwise around the curve given b the picture: 3 Work F T ds (4 2) d + (2 4) d d d 4 d d (Area) 4 2 (π(3)2 π() 2 ) 2(9π π) 6π Notation The notation P d + Q d Is sometimes used to indicate that the line integral is calculated using the positive orientation of the closed curve. 2. Another notation for the positivel oriented boundar curve of a region D is D. Fun Reads There is additional material in 6.4 that is covered in the book that MSU will not currentl be testing on. Those wishing to gain a greater understanding of the power of Green s Theorem ma wish to read the section on finding area using line integrals (top of page ) and the section on Etended Versions of Green s Theorem (starting on page ). 4
5 MTH234 hapter 6 - Vector alculus Michigan State Universit Group Work. (WW#3) Use Green s Theorem to evaluate the line integral 4 cos( ) d sin( ) d. Where is the rectangle with vertices (0, 0), (2, 0), (0, 4), and (2, 4). 2. alculate ( 4 + 2)d + (5 + sin )d where is the boundar of region shown to the right: - - 5
Green s Theorem Jeremy Orloff
Green s Theorem Jerem Orloff Line integrals and Green s theorem. Vector Fields Vector notation. In 8.4 we will mostl use the notation (v) = (a, b) for vectors. The other common notation (v) = ai + bj runs
More informationis the two-dimensional curl of the vector field F = P, Q. Suppose D is described by a x b and f(x) y g(x) (aka a type I region).
Math 55 - Vector alculus II Notes 4.4 Green s Theorem We begin with Green s Theorem: Let be a positivel oriented (parameterized counterclockwise) piecewise smooth closed simple curve in R and be the region
More informationVector fields, line integrals, and Green s Theorem
Vector fields, line integrals, and Green s Theorem Line integrals The problem: Suppose ou have a surface = f(, ) defined over a region D. Restrict the domain of the function to the values of and which
More information18.02 Review Jeremy Orloff. 1 Review of multivariable calculus (18.02) constructs
18.02 eview Jerem Orloff 1 eview of multivariable calculus (18.02) constructs 1.1 Introduction These notes are a terse summar of what we ll need from multivariable calculus. If, after reading these, some
More informationES.182A Topic 41 Notes Jeremy Orloff. 41 Extensions and applications of Green s theorem
ES.182A Topic 41 Notes Jerem Orloff 41 Etensions and applications of Green s theorem 41.1 eview of Green s theorem: Tangential (work) form: F T ds = curlf d d M d + N d = N M d d. Normal (flu) form: F
More informationExtra Problems Chapter 7
MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 1 Etra Problems hapter 7 1. onsider the vector field F = i+z j +z 3 k. a) ompute div F. b) ompute curl F. Solution a) div F = +z +3z b) curl F = i
More informationExtra Problems Chapter 7
MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 1 Etra Problems hapter 7 1. onsider the vector field F = i+z j +z 3 k. a) ompute div F. b) ompute curl F. Solution a) div F = +z +3z b) curl F = i
More information(a) We split the square up into four pieces, parametrizing and integrating one a time. Right side: C 1 is parametrized by r 1 (t) = (1, t), 0 t 1.
Thursda, November 5 Green s Theorem Green s Theorem is a 2-dimensional version of the Fundamental Theorem of alculus: it relates the (integral of) a vector field F on the boundar of a region to the integral
More informationTopic 3 Notes Jeremy Orloff
Topic 3 Notes Jerem Orloff 3 Line integrals and auch s theorem 3.1 Introduction The basic theme here is that comple line integrals will mirror much of what we ve seen for multivariable calculus line integrals.
More informationOne side of each sheet is blank and may be used as scratch paper.
Math 244 Spring 2017 (Practice) Final 5/11/2017 Time Limit: 2 hours Name: No calculators or notes are allowed. One side of each sheet is blank and may be used as scratch paper. heck your answers whenever
More informationName: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8
Name: SOLUTIONS Date: /9/7 M55 alculus III Tutorial Worksheet 8. ompute R da where R is the region bounded by x + xy + y 8 using the change of variables given by x u + v and y v. Solution: We know R is
More informationConservative fields and potential functions. (Sect. 16.3) The line integral of a vector field along a curve.
onservative fields and potential functions. (Sect. 16.3) eview: Line integral of a vector field. onservative fields. The line integral of conservative fields. Finding the potential of a conservative field.
More informationSection 17.4 Green s Theorem
Section 17.4 Green s Theorem alculating Line Integrals using ouble Integrals In the previous section, we saw an easy way to determine line integrals in the special case when a vector field F is conservative.
More informationMATH Line integrals III Fall The fundamental theorem of line integrals. In general C
MATH 255 Line integrals III Fall 216 In general 1. The fundamental theorem of line integrals v T ds depends on the curve between the starting point and the ending point. onsider two was to get from (1,
More informationNotes on Green s Theorem Northwestern, Spring 2013
Notes on Green s Theorem Northwestern, Spring 2013 The purpose of these notes is to outline some interesting uses of Green s Theorem in situations where it doesn t seem like Green s Theorem should be applicable.
More informationLecture 5. Equations of Lines and Planes. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.
Lecture 5 Equations of Lines and Planes Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 Universit of Massachusetts Februar 6, 2018 (2) Upcoming midterm eam First midterm: Wednesda Feb. 21, 7:00-9:00
More informationGreen s Theorem. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Green s Theorem
Green s Theorem MATH 311, alculus III J. obert Buchanan Department of Mathematics Fall 2011 Main Idea Main idea: the line integral around a positively oriented, simple closed curve is related to a double
More informationMTH 234 Exam 2 April 10th, Without fully opening the exam, check that you have pages 1 through 12.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in our name, etc. on this first page. Without full opening the eam, check that ou have pages 1 through 12. Show all our work on the standard response
More informationLine Integrals and Green s Theorem Jeremy Orloff
Line Integrals and Green s Theorem Jerem Orloff Vector Fields (or vector valued functions) Vector notation. In 8.4 we will mostl use the notation (v) = (a, b) for vectors. The other common notation (v)
More informationES.182A Topic 36 Notes Jeremy Orloff
ES.82A Topic 36 Notes Jerem Orloff 36 Vector fields and line integrals in the plane 36. Vector analsis We now will begin our stud of the part of 8.2 called vector analsis. This is the stud of vector fields
More informationTopic 5.5: Green s Theorem
Math 275 Notes Topic 5.5: Green s Theorem Textbook Section: 16.4 From the Toolbox (what you need from previous classes): omputing partial derivatives. Setting up and computing double integrals (this includes
More information49. Green s Theorem. The following table will help you plan your calculation accordingly. C is a simple closed loop 0 Use Green s Theorem
49. Green s Theorem Let F(x, y) = M(x, y), N(x, y) be a vector field in, and suppose is a path that starts and ends at the same point such that it does not cross itself. Such a path is called a simple
More informationSome common examples of vector fields: wind shear off an object, gravitational fields, electric and magnetic fields, etc
Vector Analysis Vector Fields Suppose a region in the plane or space is occupied by a moving fluid such as air or water. Imagine this fluid is made up of a very large number of particles that at any instant
More informationMATH Green s Theorem Fall 2016
MATH 55 Green s Theorem Fall 16 Here is a statement of Green s Theorem. It involves regions and their boundaries. In order have any hope of doing calculations, you must see the region as the set of points
More informationV11. Line Integrals in Space
V11. Line Integrals in Space 1. urves in space. In order to generalize to three-space our earlier work with line integrals in the plane, we begin by recalling the relevant facts about parametrized space
More information( 7, 3) means x = 7 and y = 3. ( 7, 3) works in both equations so. Section 5 1: Solving a System of Linear Equations by Graphing
Section 5 : Solving a Sstem of Linear Equations b Graphing What is a sstem of Linear Equations? A sstem of linear equations is a list of two or more linear equations that each represents the graph of a
More informationSOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)
SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please
More informationMultivariable Calculus
Math Spring 05 BY: $\ Ron Buckmire Multivariable alculus Worksheet 6 TITLE Path-Dependent Vector Fields and Green s Theorem URRENT READING Mcallum, Section 8.4 HW # (DUE Wednesday 04/ BY 5PM) Mcallum,
More informationLINE AND SURFACE INTEGRALS: A SUMMARY OF CALCULUS 3 UNIT 4
LINE AN URFAE INTEGRAL: A UMMARY OF ALULU 3 UNIT 4 The final unit of material in multivariable calculus introduces many unfamiliar and non-intuitive concepts in a short amount of time. This document attempts
More informationM273Q Multivariable Calculus Spring 2017 Review Problems for Exam 3
M7Q Multivariable alculus Spring 7 Review Problems for Exam Exam covers material from Sections 5.-5.4 and 6.-6. and 7.. As you prepare, note well that the Fall 6 Exam posted online did not cover exactly
More informationPractice Questions for Midterm 2 - Math 1060Q Fall
Eam Review Practice Questions for Midterm - Math 00Q - 0Fall The following is a selection of problems to help prepare ou for the second midterm eam. Please note the following: there ma be mistakes the
More informationOLD MIDTERM EXAMS AND CLASS TESTS
189-65A: Advanced alculus OLD MIDTERM EXAMS AND LASS TESTS Midterm Eam October 1996 Answer all questions in Part A. Answer two () question from Part B and one (1) question from Part for a total of si (6)
More information16.3. Conservative Vector Fields
16.3 onservative Vector Fields Review: Work F d r = FT ds = Fr '( t ) dt Mdx Nd Pdz if F Mi Nj Pk F d r is also called circulation if F represents a velocit vector field. Outward flux across a simple closed
More informationCalculus III. Math 233 Spring Final exam May 3rd. Suggested solutions
alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.
More informationLINE AND SURFACE INTEGRALS: A SUMMARY OF CALCULUS 3 UNIT 4
LINE AN URFAE INTEGRAL: A UMMARY OF ALULU 3 UNIT 4 The final unit of material in multivariable calculus introduces many unfamiliar and non-intuitive concepts in a short amount of time. This document attempts
More informationA CURL-FREE VECTOR FIELD THAT IS NOT A GRADIENT. Robert L. Foote. Math 225
A URL-FREE VETOR FIELD THAT IS NOT A GRADIENT Robert L. Foote Math 225 Recall our main theorem about vector fields. Theorem. Let R be an open region in E 2 and let F be a vector field on R. The following
More informationDirection of maximum decrease = P
APPM 35 FINAL EXAM PING 15 INTUTION: Electronic devices, books, and crib sheets are not permitted. Write your name and your instructor s name on the front of your bluebook. Work all problems. how your
More informationMath 21a: Multivariable calculus. List of Worksheets. Harvard University, Spring 2009
Math 2a: Multivariable calculus Harvard Universit, Spring 2009 List of Worksheets Vectors and the Dot Product Cross Product and Triple Product Lines and Planes Functions and Graphs Quadric Surfaces Vector-Valued
More informationArc Length and Curvature
Arc Length and Curvature. Last time, we saw that r(t) = cos t, sin t, t parameteried the pictured curve. (a) Find the arc length of the curve between (, 0, 0) and (, 0, π). (b) Find the unit tangent vector
More informationENGI 4430 Line Integrals; Green s Theorem Page 8.01
ENGI 4430 Line Integrals; Green s Theorem Page 8.01 8. Line Integrals Two applications of line integrals are treated here: the evaluation of work done on a particle as it travels along a curve in the presence
More informationSection 5-7 : Green's Theorem
Section 5-7 : Green's Theorem In this section we are going to investigate the relationship between certain kinds of line integrals (on closed paths) and double integrals. Let s start off with a simple
More informationMATH 223 FINAL EXAM STUDY GUIDE ( )
MATH 3 FINAL EXAM STUDY GUIDE (017-018) The following questions can be used as a review for Math 3 These questions are not actual samples of questions that will appear on the final eam, but the will provide
More informationScalar functions of several variables (Sect. 14.1)
Scalar functions of several variables (Sect. 14.1) Functions of several variables. On open, closed sets. Functions of two variables: Graph of the function. Level curves, contour curves. Functions of three
More informationES.182A Problem Section 11, Fall 2018 Solutions
Problem 25.1. (a) z = 2 2 + 2 (b) z = 2 2 ES.182A Problem Section 11, Fall 2018 Solutions Sketch the following quadratic surfaces. answer: The figure for part (a) (on the left) shows the z trace with =
More informationMTH 234 Solutions to Exam 2 April 10th, Without fully opening the exam, check that you have pages 1 through 12.
MTH 34 Solutions to am April 1th, 17 Name: Section: Recitation Instructor: INSTRUTIONS Fill in your name, etc. on this first page. Without fully opening the eam, check that you have pages 1 through 1.
More informationPreface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
alculus III Preface Here are my online notes for my alculus III course that I teach here at Lamar University. espite the fact that these are my class notes, they should be accessible to anyone wanting
More informationCalculus: Several Variables Lecture 27
alculus: Several Variables Lecture 27 Instructor: Maksim Maydanskiy Lecture 27 Plan 1. Work integrals over a curve continued. (15.4) Work integral and circulation. Example by inspection. omputation via
More information3, 1, 3 3, 1, 1 3, 1, 1. x(t) = t cos(πt) y(t) = t sin(πt), z(t) = t 2 t 0
Math 5 Final Eam olutions ecember 5, Problem. ( pts.) (a 5 pts.) Find the distance from the point P (,, 7) to the plane z +. olution. We can easil find a point P on the plane b choosing some values for
More informationMath 265H: Calculus III Practice Midterm II: Fall 2014
Name: Section #: Math 65H: alculus III Practice Midterm II: Fall 14 Instructions: This exam has 7 problems. The number of points awarded for each question is indicated in the problem. Answer each question
More informationMath 261 Solutions to Sample Final Exam Problems
Math 61 Solutions to Sample Final Eam Problems 1 Math 61 Solutions to Sample Final Eam Problems 1. Let F i + ( + ) j, and let G ( + ) i + ( ) j, where C 1 is the curve consisting of the circle of radius,
More informationLine Integrals and Gradient Fields
Line Integrals and Gradient Fields Based on notes by Ben Woodruff When you make your lesson plan, it should explain and contain examples of the following: 1. Describe how to integrate a function along
More informationMath 212-Lecture 20. P dx + Qdy = (Q x P y )da. C
15. Green s theorem Math 212-Lecture 2 A simple closed curve in plane is one curve, r(t) : t [a, b] such that r(a) = r(b), and there are no other intersections. The positive orientation is counterclockwise.
More informationReview Questions for Test 3 Hints and Answers
eview Questions for Test 3 Hints and Answers A. Some eview Questions on Vector Fields and Operations. A. (a) The sketch is left to the reader, but the vector field appears to swirl in a clockwise direction,
More informationReview: critical point or equivalently f a,
Review: a b f f a b f a b critical point or equivalentl f a, b A point, is called a of if,, 0 A local ma or local min must be a critical point (but not conversel) 0 D iscriminant (or Hessian) f f D f f
More informationChapter Exercise 7. Exercise 8. VectorPlot[{-x, -y}, {x, -2, 2}, {y, -2, 2}]
Chapter 5. Exercise 7 VectorPlot[{x, }, {x, -, }, {y, -, }] - - - - Exercise 8 VectorPlot[{-x, -y}, {x, -, }, {y, -, }] - - - - ch5-hw-solutions.nb Exercise VectorPlot[{y, -x}, {x, -, }, {y, -, }] - -
More informationIntegration in the Complex Plane (Zill & Wright Chapter 18)
Integration in the omplex Plane Zill & Wright hapter 18) 116-4-: omplex Variables Fall 11 ontents 1 ontour Integrals 1.1 Definition and Properties............................. 1. Evaluation.....................................
More informationFinal Exam Review Sheet : Comments and Selected Solutions
MATH 55 Applied Honors alculus III Winter Final xam Review heet : omments and elected olutions Note: The final exam will cover % among topics in chain rule, linear approximation, maximum and minimum values,
More informationLimits 4: Continuity
Limits 4: Continuit 55 Limits 4: Continuit Model : Continuit I. II. III. IV. z V. VI. z a VII. VIII. IX. Construct Your Understanding Questions (to do in class). Which is the correct value of f (a) in
More informationEXERCISES Chapter 14: Partial Derivatives. Finding Local Extrema. Finding Absolute Extrema
34 Chapter 4: Partial Derivatives EXERCISES 4.7 Finding Local Etrema Find all the local maima, local minima, and saddle points of the functions in Eercises 3... 3. 4. 5. 6. 7. 8. 9.... 3. 4. 5. 6. 7. 8.
More informationPractice problems. m zδdv. In our case, we can cancel δ and have z =
Practice problems 1. Consider a right circular cone of uniform density. The height is H. Let s say the distance of the centroid to the base is d. What is the value d/h? We can create a coordinate system
More informationMath 120: Examples. Green s theorem. x 2 + y 2 dx + x. x 2 + y 2 dy. y x 2 + y 2, Q = x. x 2 + y 2
Math 12: Examples Green s theorem Example 1. onsider the integral Evaluate it when (a) is the circle x 2 + y 2 = 1. (b) is the ellipse x 2 + y2 4 = 1. y x 2 + y 2 dx + Solution. (a) We did this in class.
More informationStokes Theorem. MATH 311, Calculus III. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Stokes Theorem
tokes Theorem MATH 311, alculus III J. Robert Buchanan Department of Mathematics ummer 2011 Background (1 of 2) Recall: Green s Theorem, M(x, y) dx + N(x, y) dy = R ( N x M ) da y where is a piecewise
More informationGreen s Theorem in the Plane
hapter 6 Green s Theorem in the Plane Recall the following special case of a general fact proved in the previous chapter. Let be a piecewise 1 plane curve, i.e., a curve in R defined by a piecewise 1 -function
More informationQuestion: Total. Points:
MATH 307 First Midterm October 6, 2010 Name: ID: Question: 1 2 3 4 5 Total Points: 16 15 15 16 12 74 Score: There are 5 problems on 6 pages in this eam (not counting the cover sheet). Make sure that ou
More informationCHAPTER 2: Partial Derivatives. 2.2 Increments and Differential
CHAPTER : Partial Derivatives.1 Definition of a Partial Derivative. Increments and Differential.3 Chain Rules.4 Local Etrema.5 Absolute Etrema 1 Chapter : Partial Derivatives.1 Definition of a Partial
More informationf x,y da 2 9. x 2 y 2 dydx y 2 dy x2 dx 2 9. y x da 4 x
MATH 3 (Calculus III) -Exam 4 (Version ) Solutions March 5, 5 S. F. Ellermeer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarit of
More informationUnit 6 Line and Surface Integrals
Unit 6 Line and Surface Integrals In this unit, we consider line integrals and surface integrals and the relationships between them. We also discuss the three theorems Green s theorem, the divergence theorem
More information4B. Line Integrals in the Plane
4. Line Integrals in the Plane 4A. Plane Vector Fields 4A-1 Describe geometrically how the vector fields determined by each of the following vector functions looks. Tell for each what the largest region
More information(6, 4, 0) = (3, 2, 0). Find the equation of the sphere that has the line segment from P to Q as a diameter.
Solutions Review for Eam #1 Math 1260 1. Consider the points P = (2, 5, 1) and Q = (4, 1, 1). (a) Find the distance from P to Q. Solution. dist(p, Q) = (4 2) 2 + (1 + 5) 2 + (1 + 1) 2 = 4 + 36 + 4 = 44
More informationFunctions of Several Variables
Chapter 1 Functions of Several Variables 1.1 Introduction A real valued function of n variables is a function f : R, where the domain is a subset of R n. So: for each ( 1,,..., n ) in, the value of f is
More information(0,2) L 1 L 2 R (-1,0) (2,0) MA4006: Exercise Sheet 3: Solutions. 1. Evaluate the integral R
MA6: Eercise Sheet 3: Solutions 1. Evaluate the integral d d over the triangle with vertices ( 1, ), (, 2) and (2, ). Solution. See Figure 1. Let be the inner variable and the outer variable. we need the
More information10.5. Polar Coordinates. 714 Chapter 10: Conic Sections and Polar Coordinates. Definition of Polar Coordinates
71 Chapter 1: Conic Sections and Polar Coordinates 1.5 Polar Coordinates rigin (pole) r P(r, ) Initial ra FIGURE 1.5 To define polar coordinates for the plane, we start with an origin, called the pole,
More informationAnswer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE
The SAT Subject Tests Answer Eplanations TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE Mathematics Level & Visit sat.org/stpractice to get more practice and stud tips for the Subject Test
More informationTriple Integrals. y x
Triple Integrals. (a) If is an solid (in space), what does the triple integral dv represent? Wh? (b) Suppose the shape of a solid object is described b the solid, and f(,, ) gives the densit of the object
More informationThis is only a list of questions use a separate sheet to work out the problems. 1. (1.2 and 1.4) Use the given graph to answer each question.
Mth Calculus Practice Eam Questions NOTE: These questions should not be taken as a complete list o possible problems. The are merel intended to be eamples o the diicult level o the regular eam questions.
More informationWorksheet #1. A little review.
Worksheet #1. A little review. I. Set up BUT DO NOT EVALUATE definite integrals for each of the following. 1. The area between the curves = 1 and = 3. Solution. The first thing we should ask ourselves
More informationName: SOLUTIONS Date: 09/07/2017. M20550 Calculus III Tutorial Worksheet 2
M20550 Calculus III Tutorial Worksheet 2 1. Find an equation of the plane passes through the point (1, 1, 7) and perpendicular to the line x = 1 + 4t, y = 1 t, z = 3. Solution: To write an equation of
More information2.5. Infinite Limits and Vertical Asymptotes. Infinite Limits
. Infinite Limits and Vertical Asmptotes. Infinite Limits and Vertical Asmptotes In this section we etend the concept of it to infinite its, which are not its as before, but rather an entirel new use of
More informationThe Divergence Theorem Stokes Theorem Applications of Vector Calculus. Calculus. Vector Calculus (III)
Calculus Vector Calculus (III) Outline 1 The Divergence Theorem 2 Stokes Theorem 3 Applications of Vector Calculus The Divergence Theorem (I) Recall that at the end of section 12.5, we had rewritten Green
More informationVector-Valued Functions
Vector-Valued Functions 1 Parametric curves 8 ' 1 6 1 4 8 1 6 4 1 ' 4 6 8 Figure 1: Which curve is a graph of a function? 1 4 6 8 1 8 1 6 4 1 ' 4 6 8 Figure : A graph of a function: = f() 8 ' 1 6 4 1 1
More information12.2. Maxima and Minima. Introduction. Prerequisites. Learning Outcomes
Maima and Minima 1. Introduction In this section we analse curves in the local neighbourhood of a stationar point and, from this analsis, deduce necessar conditions satisfied b local maima and local minima.
More information6 = 1 2. The right endpoints of the subintervals are then 2 5, 3, 7 2, 4, 2 9, 5, while the left endpoints are 2, 5 2, 3, 7 2, 4, 9 2.
5 THE ITEGRAL 5. Approimating and Computing Area Preliminar Questions. What are the right and left endpoints if [, 5] is divided into si subintervals? If the interval [, 5] is divided into si subintervals,
More informationMA 441 Advanced Engineering Mathematics I Assignments - Spring 2014
MA 441 Advanced Engineering Mathematics I Assignments - Spring 2014 Dr. E. Jacobs The main texts for this course are Calculus by James Stewart and Fundamentals of Differential Equations by Nagle, Saff
More informationMATH 280 Multivariate Calculus Fall Integrating a vector field over a curve
MATH 280 Multivariate alculus Fall 2012 Definition Integrating a vector field over a curve We are given a vector field F and an oriented curve in the domain of F as shown in the figure on the left below.
More informationInequalities and Multiplication
Lesson 3-6 Inequalities and Multiplication BIG IDEA Multipling each side of an inequalit b a positive number keeps the direction of the inequalit; multipling each side b a negative number reverses the
More informationName: Date: 12/06/2018. M20550 Calculus III Tutorial Worksheet 11
1. ompute the surface integral M255 alculus III Tutorial Worksheet 11 x + y + z) d, where is a surface given by ru, v) u + v, u v, 1 + 2u + v and u 2, v 1. olution: First, we know x + y + z) d [ ] u +
More information18.02 Multivariable Calculus Fall 2007
MIT OpenourseWare http://ocw.mit.edu 18.02 Multivariable alculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.02 Lecture 21. Test for
More informationMTH234 Chapter 15 - Multiple Integrals Michigan State University
MTH24 Chater 15 - Multile Integrals Michigan State University 6 Surface Area Just as arc length is an alication of a single integral, surface area is an alication of double integrals. In 15.6 we comute
More informationLesson 29 MA Nick Egbert
Lesson 9 MA 16 Nick Egbert Overview In this lesson we build on the previous two b complicating our domains of integration and discussing the average value of functions of two variables. Lesson So far the
More informationCopyright 2008 by Evans M. Harrell II. Going Green!
Copyright 2008 by Evans M. Harrell II. Going Green! Math stories Two mathematicians meet in the Skiles Building. The first asks the second how his family is, and the second answers: "They're great. My
More informationThe Residue Theorem. Integration Methods over Closed Curves for Functions with Singularities
The Residue Theorem Integration Methods over losed urves for Functions with Singularities We have shown that if f(z) is analytic inside and on a closed curve, then f(z)dz = 0. We have also seen examples
More informationMATHEMATICS 317 April 2017 Final Exam Solutions
MATHEMATI 7 April 7 Final Eam olutions. Let r be the vector field r = îı + ĵj + z ˆk and let r be the function r = r. Let a be the constant vector a = a îı + a ĵj + a ˆk. ompute and simplif the following
More information1. (30 points) In the x-y plane, find and classify all local maxima, local minima, and saddle points of the function. f(x, y) = 3y 2 2y 3 3x 2 + 6xy.
APPM 35 FINAL EXAM FALL 13 INSTUTIONS: Electronic devices, books, and crib sheets are not permitted. Write your name and your instructor s name on the front of your bluebook. Work all problems. Show your
More information3 Polynomial and Rational Functions
3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental
More informationMAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function
MAT 275: Introduction to Mathematical Analsis Dr. A. Rozenblum Graphs and Simplest Equations for Basic Trigonometric Functions We consider here three basic functions: sine, cosine and tangent. For them,
More informationSample Problems For Grade 9 Mathematics. Grade. 1. If x 3
Sample roblems For 9 Mathematics DIRECTIONS: This section provides sample mathematics problems for the 9 test forms. These problems are based on material included in the New York Cit curriculum for 8.
More informationENGI 4430 Line Integrals; Green s Theorem Page 8.01
ENGI 443 Line Integrals; Green s Theorem Page 8. 8. Line Integrals Two applications of line integrals are treated here: the evaluation of work done on a particle as it travels along a curve in the presence
More informationMAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.
MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant
More informationCircles in the Coordinate Plane. Find the length of each segment to the nearest tenth y. Distance Formula Square both sides.
-5 ircles in the oordinate Plane -5. Plan What You ll Learn To write an equation of a circle To find the center and radius of a circle... nd Wh To describe the position and range of three cellular telephone
More informationVector Functions & Space Curves MATH 2110Q
Vector Functions & Space Curves Vector Functions & Space Curves Vector Functions Definition A vector function or vector-valued function is a function that takes real numbers as inputs and gives vectors
More information