Unit 6 Line and Surface Integrals
|
|
- Preston Bishop
- 5 years ago
- Views:
Transcription
1 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 and Stokes s theorem that summarize the connections between line, surface, double and triple integrals. Note: Unit 6 is based on hapter 17 of the textbook, Salas and Hille s alculus: Several Variables, 7th ed., revised by Garret J. Etgen (New York: Wiley, 1995). All assigned readings and exercises are from that textbook, unless otherwise indicated. Objectives Detailed objectives are given in each of the sections listed below. 1 Line Integrals 2 The Fundamental Theorem for Line Integrals 3 Work, Energy and the onservation of Mechanical Energy 4 Alternative Notation; Line Integers and Arc Length 5 Green s Theorem 6 Parametrized Surfaces and Surface Area 7 Surface Integrals 8 The Vector Differential Operator 9 The Divergence Theorem 10 Stokes s Theorem Mathematics 365 / Study Guide 47
2 Objective 1 a. integrate a two-dimensional vector field over a path. b. integrate a three-dimensional vector field over a path. c. calculate the work done by a two-dimensional force operating over a path. d. calculate the work done by a three-dimensional force operating over a path. e. demonstrate various conclusions relating to line integrals. Read Section 17.1, pages omplete problems 1-10 and odd-numbered problems on pages scalar field vector field total work done by a force over a smooth curve: W = [ Fr ( ( u )) i r ( u )] du line integral of vector field h over curve : b hr ( ) i dr= [ hr ( ( u)) i r ( u)] du a effect of sense-preserving change of parameter on line integral piecewise smooth curve Before you proceed to Objective 2, make certain that you can meet each of the sub-objectives listed under Objective 1. b a 48 alculus Several Variables
3 Objective 2 a. use the fundamental theorem for line integrals to solve line integral problems. b. demonstrate various conclusions relating to the fundamental theorem for line integrals. Read Section 17.2, pages omplete odd-numbered problems 1-21 and problems on pages gradient field fundamental theorem for line integrals relationship between fundamental theorem for line integrals and fundamental theorem of calculus fundamental theorem for line integrals: closed curves Before you proceed to Objective 3, make certain that you can meet each of the sub-objectives listed under Objective 2. Objective 3 a. use the work-energy formula and the law of conservation of mechanical energy to solve problems relating to energy. b. demonstrate various conclusions relating to force, energy and work. Read Section 17.3, pages Mathematics 365 / Study Guide 49
4 omplete problems 1-9 on page work-energy formula W = m[()] v β m[( v α)] 2 2 conservative field 2 2 potential energy functions for a conservative force field total mechanical energy (E) law of conservation of mechanical energy mv + U = E difference in potential energy escape velocity equipotential surfaces of a conservative force field Before you proceed to Objective 4, make certain that you can meet each of the sub-objectives listed under Objective 3. Objective 4 a. use the arc length formula to evaluate a line integral along a given path, to determine whether a given vector field is a gradient field, and to find length, mass, centre of mass and moment of inertia. b. demonstrate various conclusions relating to the arc length formula. Read Section 17.4, pages omplete problems 1-29 and odd-numbered problems on pages alculus Several Variables
5 alternative notation for line integrals: Pxyzdx (,, ) + Qxyzdy (,, ) + Rxyzdz (,, ) = ( ) i hr dr definition of line integrals with respect to arc length arc length equations for length, mass, centre of mass (vector and scalar forms), moment of inertia Before you proceed to Objective 5, make certain that you can meet each of the sub-objectives listed under Objective 4. Objective 5 a. use Green s theorem to convert a line integral along a boundary of a Jordan region into a double integral, and to convert a double integral to a line integral along the boundary of a Jordan region b. use Green s theorem to evaluate line integrals, and to determine work, area and moment of inertia. c. demonstrate various conclusions relating to Green s theorem. Read Section 17.5, pages omplete odd-numbered problems 1-35 on pages Jordan curve Jordan region δq δp Green s theorem: ( x, y) ( x, y) dx dy = P( x, y) dx + Q( x, y) dy Ω δx δy Mathematics 365 / Study Guide 51
6 symbols and elementary region counterclockwise integral over 1 clockwise integral over 2 Before you proceed to Objective 6, make certain that you can meet each of the sub-objectives listed under Objective 5. Objective 6 a. calculate fundamental vector products. b. provide a parametric representation for various surfaces expressed in xyzcoordinates. c. provide a formula in xyz-coordinates for various surfaces expressed in parametric form. d. use the appropriate formula to find the area of various surfaces. e. demonstrate various conclusions relating to fundamental vector products. Read Section 17.6, pages omplete problems 1 and 3-21, and odd-numbered problems on pages parametrization of a surface (function, plane, sphere, cone, spiral ramp) fundamental vector product area of a parametrized surface continuously differentiable surface 52 alculus Several Variables
7 formulas for the area of a surface: area of S = N( u, v) du dv Ω area of 2 2 y S = [ f ( x, y)] + [ f ( x, y)] + 1 dx dy Ω x A = sec[ γ(x,y)] dx dy Ω Before you proceed to Objective 7, make certain that you can meet each of the sub-objectives listed under Objective 6. Objective 7 a. evaluate integrals over a surface. b. find the mass of a material surface. c. calculate the flux of a vector across a surface. Read Section 17.7, pages omplete problems 1-12, 18-23, 25-27, 29, 31 and 32 on pages material surface formula for the mass of a material surface: M = λ[ xuv (, ), yuv (, ), zuv (, )] Nuv (, ) dudv Ω formula for the surface integral of a scalar field continuous over a surface: H( xyz,, ) dσ= Hxuv [ (, ), yuv (, ), zuv (, )] Nuv (, ) dudv S Ω average value of a scalar field continuous over a surface G-weighted value of H on S Mathematics 365 / Study Guide 53
8 flux of a vector field across a surface in the direction of a unit normal closed piecewise-smooth function Before you proceed to Objective 8, make certain that you can meet each of the sub-objectives listed under Objective 7. Objective 8 a. find the divergence and curl of a vector field. b. calculate the Laplacian of a scalar field. c. demonstrate various conclusions relating to the vector differential operator. Read Section 17.8, pages omplete problems 1-21, 23, 25 and on pages vector differential operator : δ δ δ = i + j + k δx δy δz gradient of f: f f f f δ δ δ = i + j + k f = δ + δ + δ δx δy δz i j k δx δy δz δv1 δv2 δv3 divergence of v: i v = + + δx δy δz curl of v: i j k δ δ δ δv3 δv2 δv1 δv3 δv2 δv1 v = = x y z y z i + j+ δ δ δ δ δ δz δx δx δy k v v v alculus Several Variables
9 Laplacian operator: δ f δ f δ f f = i ( f ) = δx δy δz basic identities: the curl of a gradient is zero the divergence of a curl is zero product rule for divergence product rule for curl n i ( r r) = ( n + 3) r ( r n r)= 0 n Before you proceed to Objective 9, make certain that you can meet each of the sub-objectives listed under Objective 8. Objective 9 a. use the divergence theorem to calculate the flux out of a solid. b. demonstrate various conclusions relating to the divergence theorem. Read Section 17.9, pages omplete odd-numbered problems 1-19 on pages Green s theorem expressed in vector terms: ( i v) dx dy = ( v i n) ds divergence theorem (Gauss s theorem) as a higher-dimensional analogue of Green s theorem: ( i v) dx dy dz = ( v i n) dσ T divergence as outward flux per unit volume S Ω Mathematics 365 / Study Guide 55
10 positive divergence source sink solenoidal electric field Before you proceed to Objective 10, make certain that you can meet each of the sub-objectives listed under Objective 9. Objective 10 a. verify Stokes s theorem for particular examples of smooth surfaces with smooth bounding curves. b. solve problems using Stokes s theorem. c. demonstrate various conclusions relating to Stokes s theorem. Read Section 17.10, pages omplete odd-numbered problems 1-11, and problems on pages Stokes s theorem: [( v) i n] dσ= v( r) i dr circulation per unit area irrotational S Before you complete the fourth and fifth tutor-marked assignments, and write the final examination, make certain that you can meet each of the subobjectives listed under Objective alculus Several Variables
11 Final Examination Before you begin the fourth tutor-marked assignment, contact the Office of the Registrar to request the final examination. Please see your Student Manual for further information. Assignments 3 and 4 omplete Tutor-marked Assignment 3 and Tutor-marked Assignment 4, both of which you will find in the Assignments for redit section of your Student Manual. Submit the completed assignments to your tutor for grading. Remember to include a Tutor-marked Exercise form, from the course package, with each assignment, and to keep a copy of your work, at least a rough draft, in case the original is lost in the mail. ourse Project omplete the ourse Project, which you will find in the Assignments for redit section of your Student Manual. Submit the completed assignment to your tutor for grading. Remember to include a Tutor-marked Exercise form, from the course package, with each assignment, and to keep a copy of your work, at least a rough draft, in case the original is lost in the mail. Mathematics 365 / Study Guide 57
12 58 alculus Several Variables
Unit 2 Vector Calculus
Unit 2 Vector Calculus In this unit, we consider vector functions, differentiation formulas, curves, arc length, curvilinear motion, vector calculus in mechanics, planetary motion and curvature. Note:
More informationPRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.
PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x
More informationMultiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015
Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction
More informationVector Calculus. Dr. D. Sukumar. February 1, 2016
Vector Calculus Dr. D. Sukumar February 1, 2016 Green s Theorem Tangent form or Ciculation-Curl form c Mdx + Ndy = R ( N x M ) da y Green s Theorem Tangent form or Ciculation-Curl form Stoke s Theorem
More informationDetailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors
Unit 1 Vectors In this unit, we introduce vectors, vector operations, and equations of lines and planes. Note: Unit 1 is based on Chapter 12 of the textbook, Salas and Hille s Calculus: Several Variables,
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 informationMATH 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
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 informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)
More informationClass 4 : Maxwell s Equations for Electrostatics
Class 4 : Maxwell s Equations for Electrostatics Concept of charge density Maxwell s 1 st and 2 nd Equations Physical interpretation of divergence and curl How do we check whether a given vector field
More informationChapter 3 - Vector Calculus
Chapter 3 - Vector Calculus Gradient in Cartesian coordinate system f ( x, y, z,...) dr ( dx, dy, dz,...) Then, f f f f,,,... x y z f f f df dx dy dz... f dr x y z df 0 (constant f contour) f dr 0 or f
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 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 informationIntroduction and Vectors Lecture 1
1 Introduction Introduction and Vectors Lecture 1 This is a course on classical Electromagnetism. It is the foundation for more advanced courses in modern physics. All physics of the modern era, from quantum
More informationVector Calculus, Maths II
Section A Vector Calculus, Maths II REVISION (VECTORS) 1. Position vector of a point P(x, y, z) is given as + y and its magnitude by 2. The scalar components of a vector are its direction ratios, and represent
More informationPast Exam Problems in Integrals, Solutions
Past Exam Problems in Integrals, olutions Prof. Qiao Zhang ourse 11.22 December 7, 24 Note: These problems do not imply, in any sense, my taste or preference for our own exam. ome of the problems here
More informationCURRENT MATERIAL: Vector Calculus.
Math 275, section 002 (Ultman) Fall 2011 FINAL EXAM REVIEW The final exam will be held on Wednesday 14 December from 10:30am 12:30pm in our regular classroom. You will be allowed both sides of an 8.5 11
More informationVector Calculus. Dr. D. Sukumar. January 31, 2014
Vector Calculus Dr. D. Sukumar January 31, 2014 Green s Theorem Tangent form or Ciculation-Curl form c Mdx +Ndy = R ( N x M ) da y Green s Theorem Tangent form or Ciculation-Curl form c Mdx +Ndy = C F
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 informationMATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS
MATH 228: Calculus III (FALL 216) Sample Problems for FINAL EXAM SOLUTIONS MATH 228 Page 2 Problem 1. (2pts) Evaluate the line integral C xy dx + (x + y) dy along the parabola y x2 from ( 1, 1) to (2,
More informationDO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START
Math 265 Student name: KEY Final Exam Fall 23 Instructor & Section: This test is closed book and closed notes. A (graphing) calculator is allowed for this test but cannot also be a communication device
More informationSept , 17, 23, 29, 37, 41, 45, 47, , 5, 13, 17, 19, 29, 33. Exam Sept 26. Covers Sept 30-Oct 4.
MATH 23, FALL 2013 Text: Calculus, Early Transcendentals or Multivariable Calculus, 7th edition, Stewart, Brooks/Cole. We will cover chapters 12 through 16, so the multivariable volume will be fine. WebAssign
More informationMa 1c Practical - Solutions to Homework Set 7
Ma 1c Practical - olutions to omework et 7 All exercises are from the Vector Calculus text, Marsden and Tromba (Fifth Edition) Exercise 7.4.. Find the area of the portion of the unit sphere that is cut
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 informationLecture II: Vector and Multivariate Calculus
Lecture II: Vector and Multivariate Calculus Dot Product a, b R ' ', a ( b = +,- a + ( b + R. a ( b = a b cos θ. θ convex angle between the vectors. Squared norm of vector: a 3 = a ( a. Alternative notation:
More informationFinal Exam. Monday March 19, 3:30-5:30pm MAT 21D, Temple, Winter 2018
Name: Student ID#: Section: Final Exam Monday March 19, 3:30-5:30pm MAT 21D, Temple, Winter 2018 Show your work on every problem. orrect answers with no supporting work will not receive full credit. Be
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 informationChapter 1. Vector Algebra and Vector Space
1. Vector Algebra 1.1. Scalars and vectors Chapter 1. Vector Algebra and Vector Space The simplest kind of physical quantity is one that can be completely specified by its magnitude, a single number, together
More informationMATH 2400: Calculus III, Fall 2013 FINAL EXAM
MATH 2400: Calculus III, Fall 2013 FINAL EXAM December 16, 2013 YOUR NAME: Circle Your Section 001 E. Angel...................... (9am) 002 E. Angel..................... (10am) 003 A. Nita.......................
More informationNotes 3 Review of Vector Calculus
ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018 A ˆ Notes 3 Review of Vector Calculus y ya ˆ y x xa V = x y ˆ x Adapted from notes by Prof. Stuart A. Long 1 Overview Here we present
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 informationPractice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.
1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line
More informationPhysics 6303 Lecture 2 August 22, 2018
Physics 6303 Lecture 2 August 22, 2018 LAST TIME: Coordinate system construction, covariant and contravariant vector components, basics vector review, gradient, divergence, curl, and Laplacian operators
More informationBrief Review of Vector Algebra
APPENDIX Brief Review of Vector Algebra A.0 Introduction Vector algebra is used extensively in computational mechanics. The student must thus understand the concepts associated with this subject. The current
More informationMathematical Concepts & Notation
Mathematical Concepts & Notation Appendix A: Notation x, δx: a small change in x t : the partial derivative with respect to t holding the other variables fixed d : the time derivative of a quantity that
More informationMATH 52 FINAL EXAM DECEMBER 7, 2009
MATH 52 FINAL EXAM DECEMBER 7, 2009 THIS IS A CLOSED BOOK, CLOSED NOTES EXAM. NO CALCULATORS OR OTHER ELECTRONIC DEVICES ARE PERMITTED. IF YOU NEED EXTRA SPACE, PLEASE USE THE BACK OF THE PREVIOUS PROB-
More informationMath 302 Outcome Statements Winter 2013
Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a
More information1.1. Fields Partial derivatives
1.1. Fields A field associates a physical quantity with a position A field can be also time dependent, for example. The simplest case is a scalar field, where given physical quantity can be described by
More informationVector Calculus. A primer
Vector Calculus A primer Functions of Several Variables A single function of several variables: f: R $ R, f x (, x ),, x $ = y. Partial derivative vector, or gradient, is a vector: f = y,, y x ( x $ Multi-Valued
More informationMATH Harrell. An integral workout. Lecture 21. Copyright 2013 by Evans M. Harrell II.
MATH 2411 - Harrell An integral workout Lecture 21 Copyright 2013 by Evans M. Harrell II. This week s learning plan and announcements We ll integrate over curves Then we ll integrate over... curved surfaces
More informationSummary for Vector Calculus and Complex Calculus (Math 321) By Lei Li
Summary for Vector alculus and omplex alculus (Math 321) By Lei Li 1 Vector alculus 1.1 Parametrization urves, surfaces, or volumes can be parametrized. Below, I ll talk about 3D case. Suppose we use e
More informationUL XM522 Mutivariable Integral Calculus
1 UL XM522 Mutivariable Integral Calculus Instructor: Margarita Kanarsky 2 3 Vector fields: Examples: inverse-square fields the vector field for the gravitational force 4 The Gradient Field: 5 The Divergence
More informationJEFFERSON COLLEGE COURSE SYLLABUS MTH201 CALCULUS III. 5 Semester Credit Hours. Prepared by: Linda Cook
JEFFERSON COLLEGE COURSE SYLLABUS MTH201 CALCULUS III 5 Semester Credit Hours Prepared by: Linda Cook Revised Date: December 14, 2006 by Mulavana J Johny Arts & Science Education Dr. Mindy Selsor, Dean
More informationIntroduction to Vector Calculus (29) SOLVED EXAMPLES. (d) B. C A. (f) a unit vector perpendicular to both B. = ˆ 2k = = 8 = = 8
Introduction to Vector Calculus (9) SOLVED EXAMPLES Q. If vector A i ˆ ˆj k, ˆ B i ˆ ˆj, C i ˆ 3j ˆ kˆ (a) A B (e) A B C (g) Solution: (b) A B (c) A. B C (d) B. C A then find (f) a unit vector perpendicular
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 informationDivergence Theorem and Its Application in Characterizing
Divergence Theorem and Its Application in Characterizing Fluid Flow Let v be the velocity of flow of a fluid element and ρ(x, y, z, t) be the mass density of fluid at a point (x, y, z) at time t. Thus,
More informationSections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed.
MTH 34 Review for Exam 4 ections 16.1-16.8. 5 minutes. 5 to 1 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. Review for Exam 4 (16.1) Line
More informationMultiple Integrals and Vector Calculus: Synopsis
Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors - recap of basic principles. Things which are (and are not) vectors. Differentiation and integration
More informationMath 11 Fall 2018 Practice Final Exam
Math 11 Fall 218 Practice Final Exam Disclaimer: This practice exam should give you an idea of the sort of questions we may ask on the actual exam. Since the practice exam (like the real exam) is not long
More informationMajor Ideas in Calc 3 / Exam Review Topics
Major Ideas in Calc 3 / Exam Review Topics Here are some highlights of the things you should know to succeed in this class. I can not guarantee that this list is exhaustive!!!! Please be sure you are able
More informationMath Divergence and Curl
Math 23 - Divergence and Curl Peter A. Perry University of Kentucky November 3, 28 Homework Work on Stewart problems for 6.5: - (odd), 2, 3-7 (odd), 2, 23, 25 Finish Homework D2 due tonight Begin Homework
More informationDisclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.
Disclaimer: This is meant to help you start studying. It is not necessarily a complete list of everything you need to know. The MTH 234 final exam mainly consists of standard response questions where students
More informationG G. G. x = u cos v, y = f(u), z = u sin v. H. x = u + v, y = v, z = u v. 1 + g 2 x + g 2 y du dv
1. Matching. Fill in the appropriate letter. 1. ds for a surface z = g(x, y) A. r u r v du dv 2. ds for a surface r(u, v) B. r u r v du dv 3. ds for any surface C. G x G z, G y G z, 1 4. Unit normal N
More informationCURRENT MATERIAL: Vector Calculus.
Math 275, section 002 (Ultman) Spring 2012 FINAL EXAM REVIEW The final exam will be held on Wednesday 9 May from 8:00 10:00am in our regular classroom. You will be allowed both sides of two 8.5 11 sheets
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 informationPrint Your Name: Your Section:
Print Your Name: Your Section: Mathematics 1c. Practice Final Solutions This exam has ten questions. J. Marsden You may take four hours; there is no credit for overtime work No aids (including notes, books,
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 informationMATH 311 Topics in Applied Mathematics I Lecture 36: Surface integrals (continued). Gauss theorem. Stokes theorem.
MATH 311 Topics in Applied Mathematics I Lecture 36: Surface integrals (continued). Gauss theorem. Stokes theorem. Surface integrals Let X : R 3 be a smooth parametrized surface, where R 2 is a bounded
More informationTangent Planes, Linear Approximations and Differentiability
Jim Lambers MAT 80 Spring Semester 009-10 Lecture 5 Notes These notes correspond to Section 114 in Stewart and Section 3 in Marsden and Tromba Tangent Planes, Linear Approximations and Differentiability
More informationEE2007: Engineering Mathematics II Vector Calculus
EE2007: Engineering Mathematics II Vector Calculus Ling KV School of EEE, NTU ekvling@ntu.edu.sg Rm: S2-B2b-22 Ver 1.1: Ling KV, October 22, 2006 Ver 1.0: Ling KV, Jul 2005 EE2007/Ling KV/Aug 2006 EE2007:
More informationMATH Calculus IV Spring 2014 Three Versions of the Divergence Theorem
MATH 2443 008 Calculus IV pring 2014 Three Versions of the Divergence Theorem In this note we will establish versions of the Divergence Theorem which enable us to give it formulations of div, grad, and
More informationFinal exam (practice 1) UCLA: Math 32B, Spring 2018
Instructor: Noah White Date: Final exam (practice 1) UCLA: Math 32B, Spring 218 This exam has 7 questions, for a total of 8 points. Please print your working and answers neatly. Write your solutions in
More informationMath 234 Exam 3 Review Sheet
Math 234 Exam 3 Review Sheet Jim Brunner LIST OF TOPIS TO KNOW Vector Fields lairaut s Theorem & onservative Vector Fields url Divergence Area & Volume Integrals Using oordinate Transforms hanging the
More informationMathematical Notation Math Calculus & Analytic Geometry III
Mathematical Notation Math 221 - alculus & Analytic Geometry III Use Word or WordPerect to recreate the ollowing documents. Each article is worth 10 points and should be emailed to the instructor at james@richland.edu.
More informationVector and Tensor Calculus
Appendices 58 A Vector and Tensor Calculus In relativistic theory one often encounters vector and tensor expressions in both three- and four-dimensional form. The most important of these expressions are
More informationCourse Outline. 2. Vectors in V 3.
1. Vectors in V 2. Course Outline a. Vectors and scalars. The magnitude and direction of a vector. The zero vector. b. Graphical vector algebra. c. Vectors in component form. Vector algebra with components.
More informationFundamental Theorems of Vector
Chapter 17 Analysis Fundamental Theorems of Vector Useful Tip: If you are reading the electronic version of this publication formatted as a Mathematica Notebook, then it is possible to view 3-D plots generated
More informationMATH H53 : Final exam
MATH H53 : Final exam 11 May, 18 Name: You have 18 minutes to answer the questions. Use of calculators or any electronic items is not permitted. Answer the questions in the space provided. If you run out
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 informationStokesI Theorem. Definition. Let F = PT + QT + $ be a continuously differentiable,
StokesI Theorem Our text states and proves Stokes' Theorem in 12.11, but ituses the scalar form for writing both the line integral and the surface integral involved. In the applications, it is the vector
More informationCorrections to the First Printing: Chapter 6. This list includes corrections and clarifications through November 6, 1999.
June 2,2 1 Corrections to the First Printing: Chapter 6 This list includes corrections and clarifications through November 6, 1999. We should have mentioned that our treatment of differential forms, especially
More informationSOME PROBLEMS YOU SHOULD BE ABLE TO DO
OME PROBLEM YOU HOULD BE ABLE TO DO I ve attempted to make a list of the main calculations you should be ready for on the exam, and included a handful of the more important formulas. There are no examples
More informationMULTIVARIABLE INTEGRATION
MULTIVARIABLE INTEGRATION (SPHERICAL POLAR COORDINATES) Question 1 a) Determine with the aid of a diagram an expression for the volume element in r, θ, ϕ. spherical polar coordinates, ( ) [You may not
More informationMathematics (Course B) Lent Term 2005 Examples Sheet 2
N12d Natural Sciences, Part IA Dr M. G. Worster Mathematics (Course B) Lent Term 2005 Examples Sheet 2 Please communicate any errors in this sheet to Dr Worster at M.G.Worster@damtp.cam.ac.uk. Note that
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 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 informationSummary of various integrals
ummary of various integrals Here s an arbitrary compilation of information about integrals Moisés made on a cold ecember night. 1 General things o not mix scalars and vectors! In particular ome integrals
More information52. The Del Operator: Divergence and Curl
52. The Del Operator: Divergence and Curl Let F(x, y, z) = M(x, y, z), N(x, y, z), P(x, y, z) be a vector field in R 3. The del operator is represented by the symbol, and is written = x, y, z, or = x,
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 informationGreen s, Divergence, Stokes: Statements and First Applications
Math 425 Notes 12: Green s, Divergence, tokes: tatements and First Applications The Theorems Theorem 1 (Divergence (planar version)). Let F be a vector field in the plane. Let be a nice region of the plane
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 30. Tue, Nov
More informationMcGill University April 16, Advanced Calculus for Engineers
McGill University April 16, 2014 Faculty of cience Final examination Advanced Calculus for Engineers Math 264 April 16, 2014 Time: 6PM-9PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer
More informationDivergence Theorem December 2013
Divergence Theorem 17.3 11 December 2013 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:
More informationMathematical Notation Math Calculus & Analytic Geometry III
Name : Mathematical Notation Math 221 - alculus & Analytic Geometry III Use Word or WordPerect to recreate the ollowing documents. Each article is worth 10 points and can e printed and given to the instructor
More informationMathematical Tripos Part IA Lent Term Example Sheet 1. Calculate its tangent vector dr/du at each point and hence find its total length.
Mathematical Tripos Part IA Lent Term 205 ector Calculus Prof B C Allanach Example Sheet Sketch the curve in the plane given parametrically by r(u) = ( x(u), y(u) ) = ( a cos 3 u, a sin 3 u ) with 0 u
More informationENGI Gradient, Divergence, Curl Page 5.01
ENGI 94 5. - Gradient, Divergence, Curl Page 5. 5. The Gradient Operator A brief review is provided here for the gradient operator in both Cartesian and orthogonal non-cartesian coordinate systems. Sections
More informationMATHS 267 Answers to Stokes Practice Dr. Jones
MATH 267 Answers to tokes Practice Dr. Jones 1. Calculate the flux F d where is the hemisphere x2 + y 2 + z 2 1, z > and F (xz + e y2, yz, z 2 + 1). Note: the surface is open (doesn t include any of the
More informationExercise 1 (Formula for connection 1-forms) Using the first structure equation, show that
1 Stokes s Theorem Let D R 2 be a connected compact smooth domain, so that D is a smooth embedded circle. Given a smooth function f : D R, define fdx dy fdxdy, D where the left-hand side is the integral
More informationCOURSE OUTLINE. Course Number Course Title Credits MAT251 Calculus III 4
COURSE OUTLINE Course Number Course Title Credits MAT251 Calculus III 4 Hours: Lecture/Lab/Other 4 Lecture Co- or Pre-requisite MAT152 with a minimum C grade or better, successful completion of an equivalent
More information9.7 Gradient of a Scalar Field. Directional Derivative. Mean Value Theorem. Special Cases
SEC. 9.7 Gradient of a Scalar Field. Directional Derivative 395 Mean Value Theorems THEOREM Mean Value Theorem Let f(x, y, z) be continuous and have continuous first partial derivatives in a domain D in
More informationCOWLEY COLLEGE & Area Vocational Technical School
COWLEY COLLEGE & Area Vocational Technical School COURSE PROCEDURE FOR Student Level: This course is open to students on the college level in the sophomore year. Prerequisite: Minimum grade of C in MATH
More informationARNOLD PIZER rochester problib from CVS Summer 2003
ARNOLD PIZER rochester problib from VS Summer 003 WeBWorK assignment Vectoralculus due 5/3/08 at :00 AM.( pt) setvectoralculus/ur V.pg onsider the transformation T : x 8 53 u 45 45 53v y 53 u 8 53 v A.
More informationDivergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem
Divergence Theorem 17.3 11 December 213 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:
More informationMATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am-12:00pm.
MATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am-12:00pm. Bring a calculator and something to write with. Also, you will be allowed to bring in one 8.5 11 sheet of paper
More informationENGI Duffing s Equation Page 4.65
ENGI 940 4. - Duffing s Equation Page 4.65 4. Duffing s Equation Among the simplest models of damped non-linear forced oscillations of a mechanical or electrical system with a cubic stiffness term is Duffing
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 informationFundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics
Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/
More informationDepartment of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008
Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008 Code: C-031 Date and time: 17 Nov, 2008, 9:30 A.M. - 12:30 P.M. Maximum Marks: 45 Important Instructions: 1. The question
More informationMath 11 Fall 2007 Practice Problem Solutions
Math 11 Fall 27 Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,
More informationENGI Gradient, Divergence, Curl Page 5.01
ENGI 940 5.0 - Gradient, Divergence, Curl Page 5.0 5. e Gradient Operator A brief review is provided ere for te gradient operator in bot Cartesian and ortogonal non-cartesian coordinate systems. Sections
More information