F ds, where F and S are as given.
|
|
- Jonathan Shaw
- 6 years ago
- Views:
Transcription
1 Math 21a Integral Theorems Review pring, 29 1 For these problems, find F dr, where F and are as given. a) F x, y, z and is parameterized by rt) t, t, t t 1) b) F x, y, z and is parameterized by rt) t, t, 3 t t 1) c) F 2xy, x 2 + z, y + 2z and is parameterized by rt) t 2 t, sinπt), cos 2 πt) t 1) d) F y, x 2 +y 2 x and is the unit circle in the xy-plane, oriented counter-clockwise. x 2 +y 2 2 For these problems, find F d, where F and are as given. a) F curl 2y, z 2x, yz and is the hemisphere of radius 1, centered at the origin and above the xy-plane, oriented with the upward-pointing normal. b) F curl 2y, z 2x, yz and is the solid disk of radius 1 in the xy-plane, centered at the origin, oriented with the upward-pointing normal. c) F curl 2y, z 2x, yz and a) b) ) is the union of the two surfaces from parts a) and b), oriented with the outward-pointing normals. d) F x 2 y 3x, xy cosy)z, siny)z 2 and a) b) ) is the same as in part c). 3 For these problems, find div F dv, where F and are as given. a) F curl G where G is any appropriately smooth vector field) and is any simple solid b) F x, y 2, 2yz and is the solid ball of radius a centered at the origin c) F x 2, 2yz, x 2 z 2 and is the unit cube with corners at,, ) and 1, 1, 1)
2 Integral Theorems Review Answers and olutions 1 a) To compute this we just do it. Using rt) t, t, t, we get Frt)) t, t, t and dr 1, 1, 1 dt, so 1 1 F dr t, t, t 1, 1, 1 dt 3t dt 3 2 b) Here we could just do it again, but the parameterization is messier. It s simpler to notice that the vector field F x, y, z is conservative and thus independent of path. We see this by noticing that i j k curl F x y z x y z or simply that F f where f 1 2 x2 + y 2 + z 2 ). ince the path in part b) starts and ends at the same places as the path in part a), we can simply integrate over that path. This is what we ve done in part a), so we get the same answer: 3. 2 Another approach would be to use the fundamental theorem of line integrals: f dr fr1)) fr)) f1, 1, 1) f,, ) as before ) 3 2, c) Here the key is that the given vector field is conservative. We can see this by computing curl F or writing F f, where f x 2 y + yz + z 2. ince is a closed curve one that starts and ends at the point,, 1)), the integral must be zero. We could also see this via the fundamental theorem of line integrals: f dr fr1)) fr)) f,, 1) f,, 1). It s also possible to see this using tokes theorem where is some any!) oriented surface with as its boundary): F dr curl F d d. ither approach is fine. d) This is a cautionary tale. Let s think of our vector as F P, Q. Then it s straightforward to see that Q x P y y2 x 2 x 2 + y 2 ), 2 so we might think that we can proceed as in part c). That is, we could say that F is conservative and thus F dr. Or we might apply Green s theorem: let D be the unit disk with as it s boundary), so Q F dr x P ) da da. y D Both these conclusions are WRONG! and in fact the line integral has value 2π. D
3 What has gone wrong? We ve tried to blindly apply theorems without checking the hypotheses. In the first case we tried to apply theorem 6 on page 928) that says that F is conservative when P y Q x. But we must have this equality on a simply connected region D. In our case, both P and Q are undefined at the origin, ), so any region D containing the unit circle also contains a hole at the origin. imilarly, Green s theorem require that P and Q have continuous derivatives inside D, which again fails at the origin. We can compute the actual value using a simple parameterization of : rt) x, y cost), sint) and so drt) sint), cost) dt. Thus as claimed. F dr sint), cost) 1 1 sin 2 t) + cos 2 t) ) dt 1 dt 2π, sint), cost) dt 2 a) While we could compute this directly, it seems easier to use tokes theorem to compute a line integral instead. Here the boundary is simply the unit circle in the xy-plane, so we can parameterize as rt) x, y, z cost), sint),. Thus the vector we are integrating is 2y, z 2x, yz 2 sint), 2 cost), while dr sint), cost), dt. Thus curl 2y, z 2x, yz d 2y, z 2x, yz dr 2 sint), 2 cost), sint), cost), dt 2 sin 2 t) 2 cos 2 t) ) dt 2 dt 2t 2π 4π. b) As with part a), we could compute this integral directly. Now, however, we have even more of an incentive to use tokes theorem and compute the line integral we ve already computed this line integral! That is, our surface has the same boundary the unit circle in the xy-plane) as the surface from part a), so we can simply use our answer from there: curl 2y, z 2x, yz d 2y, z 2x, yz dr 4π.
4 curl 2y, z 2x, yz d + c) Our surface is a) b), the union of the surface from part a) with the same outwardpointing normal) and the surface from part b) with the downward-pointing normal, the opposite orientation from part b)). Thus curl 2y, z 2x, yz d curl 2y, z 2x, yz d a) b) curl 2y, z 2x, yz d curl 2y, z 2x, yz d a) 4π) 4π). b) Notice that it doesn t matter what the values of the surface integrals from part a) and part b) were, it only matters that they were the same. Thus the given surface integral over the closed surface a) b) will always be zero. This is because of two facts: i) Both a) and b) are oriented surfaces that have the curve as oriented) as their boundary, and ii) The flux integrals over a) and b) involve integrating a vector field that is a curl, and thus we can apply tokes theorem. Another approach is to apply the Divergence theorem. That is, since we re integrating over a closed surface that bounds a solid hemisphere we ll call this ), then curl 2y, z 2x, yz d div curl 2y, z 2x, yz ) dv. Now the key is the divcurl G) for any vector field G, so this integral is zero. d) This is very similar to part c). Perhaps the simplest approach from c) that we could use here is the last one, using the Divergence theorem. In this approach we again let be the solid unit hemisphere that has as its oriented) boundary. Thus F d div F) dv, where F x 2 y 3x, xy cosy)z, siny)z 2 so div F x 2 y 3x ) + xy cosy)z ) + ) siny)z 2 x y z 2xy + 2xy 2 siny)z) + 2 siny)z 3. Thus our flux integral is again!) zero. ) 2 F d 3) dv 3Vol) 3 3 π 2π, where we ve used the fact that the volume of a hemisphere is 2 3 πr3 and here r 1).
5 3 a) The key here is that div curl G) this is just an application of lairaut s theorem on order of partial derivatives). Thus div F dv div curl G) dv dv. b) Here Thus div F x x) + y y2 ) + 2yz) 1 + 2y 2y 1. z div F dv 1 dv Vol) 4 3 πa3, since is a ball of radius a. c) Here div F x x2 ) + y 2yz) + z x2 z 2 ) 2x + 2z 2z 2x. Thus, since {x, y, z) : x 1, y 1, z 1}, we get div F dv 2x dv x dx dy dz 1.
The Divergence Theorem
Math 1a The Divergence Theorem 1. Parameterize the boundary of each of the following with positive orientation. (a) The solid x + 4y + 9z 36. (b) The solid x + y z 9. (c) The solid consisting of all points
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 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 informationMath 233. Practice Problems Chapter 15. i j k
Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed
More informationMath 11 Fall 2016 Final Practice Problem Solutions
Math 11 Fall 216 Final 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 informationMath 263 Final. (b) The cross product is. i j k c. =< c 1, 1, 1 >
Math 63 Final Problem 1: [ points, 5 points to each part] Given the points P : (1, 1, 1), Q : (1,, ), R : (,, c 1), where c is a parameter, find (a) the vector equation of the line through P and Q. (b)
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 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 informationSection 6-5 : Stokes' Theorem
ection 6-5 : tokes' Theorem In this section we are going to take a look at a theorem that is a higher dimensional version of Green s Theorem. In Green s Theorem we related a line integral to a double integral
More informationHOMEWORK 8 SOLUTIONS
HOMEWOK 8 OLUTION. Let and φ = xdy dz + ydz dx + zdx dy. let be the disk at height given by: : x + y, z =, let X be the region in 3 bounded by the cone and the disk. We orient X via dx dy dz, then by definition
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 informationMATH 2400 Final Exam Review Solutions
MATH Final Eam eview olutions. Find an equation for the collection of points that are equidistant to A, 5, ) and B6,, ). AP BP + ) + y 5) + z ) 6) y ) + z + ) + + + y y + 5 + z 6z + 9 + 6 + y y + + z +
More informationSolutions for the Practice Final - Math 23B, 2016
olutions for the Practice Final - Math B, 6 a. True. The area of a surface is given by the expression d, and since we have a parametrization φ x, y x, y, f x, y with φ, this expands as d T x T y da xy
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 information(a) 0 (b) 1/4 (c) 1/3 (d) 1/2 (e) 2/3 (f) 3/4 (g) 1 (h) 4/3
Math 114 Practice Problems for Test 3 omments: 0. urface integrals, tokes Theorem and Gauss Theorem used to be in the Math40 syllabus until last year, so we will look at some of the questions from those
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 informationSolutions to Sample Questions for Final Exam
olutions to ample Questions for Final Exam Find the points on the surface xy z 3 that are closest to the origin. We use the method of Lagrange Multipliers, with f(x, y, z) x + y + z for the square of the
More informationMath 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin
Math 45 Homework et olutions Points. ( pts) The integral is, x + z y d = x + + z da 8 6 6 where is = x + z 8 x + z = 4 o, is the disk of radius centered on the origin. onverting to polar coordinates then
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 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 234 Final Exam (with answers) Spring 2017
Math 234 Final Exam (with answers) pring 217 1. onsider the points A = (1, 2, 3), B = (1, 2, 2), and = (2, 1, 4). (a) [6 points] Find the area of the triangle formed by A, B, and. olution: One way to solve
More information18.1. Math 1920 November 29, ) Solution: In this function P = x 2 y and Q = 0, therefore Q. Converting to polar coordinates, this gives I =
Homework 1 elected olutions Math 19 November 9, 18 18.1 5) olution: In this function P = x y and Q =, therefore Q x P = x. We obtain the following integral: ( Q I = x ydx = x P ) da = x da. onverting to
More informationMAT 211 Final Exam. Spring Jennings. Show your work!
MAT 211 Final Exam. pring 215. Jennings. how your work! Hessian D = f xx f yy (f xy ) 2 (for optimization). Polar coordinates x = r cos(θ), y = r sin(θ), da = r dr dθ. ylindrical coordinates x = r cos(θ),
More informationAssignment 11 Solutions
. Evaluate Math 9 Assignment olutions F n d, where F bxy,bx y,(x + y z and is the closed surface bounding the region consisting of the solid cylinder x + y a and z b. olution This is a problem for which
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 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 informationVector Calculus handout
Vector Calculus handout The Fundamental Theorem of Line Integrals Theorem 1 (The Fundamental Theorem of Line Integrals). Let C be a smooth curve given by a vector function r(t), where a t b, and let f
More informationJim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt
Jim Lambers MAT 28 ummer emester 212-1 Practice Final Exam olution 1. Evaluate the line integral xy dx + e y dy + xz dz, where is given by r(t) t 4, t 2, t, t 1. olution From r (t) 4t, 2t, t 2, we obtain
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 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 (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 informationS12.1 SOLUTIONS TO PROBLEMS 12 (ODD NUMBERS)
OLUTION TO PROBLEM 2 (ODD NUMBER) 2. The electric field is E = φ = 2xi + 2y j and at (2, ) E = 4i + 2j. Thus E = 2 5 and its direction is 2i + j. At ( 3, 2), φ = 6i + 4 j. Thus the direction of most rapid
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 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 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 informationIntegral Theorems. September 14, We begin by recalling the Fundamental Theorem of Calculus, that the integral is the inverse of the derivative,
Integral Theorems eptember 14, 215 1 Integral of the gradient We begin by recalling the Fundamental Theorem of Calculus, that the integral is the inverse of the derivative, F (b F (a f (x provided f (x
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 information7a3 2. (c) πa 3 (d) πa 3 (e) πa3
1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin
More informationProblem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j (0 t 2π). Solution: We assume a > b > 0.
MATH 64: FINAL EXAM olutions Problem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j ( t π). olution: We assume a > b >. A = 1 π (xy yx )dt = 3ab π
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 informationReview problems for the final exam Calculus III Fall 2003
Review problems for the final exam alculus III Fall 2003 1. Perform the operations indicated with F (t) = 2t ı 5 j + t 2 k, G(t) = (1 t) ı + 1 t k, H(t) = sin(t) ı + e t j a) F (t) G(t) b) F (t) [ H(t)
More informationMath Review for Exam 3
1. ompute oln: (8x + 36xy)ds = Math 235 - Review for Exam 3 (8x + 36xy)ds, where c(t) = (t, t 2, t 3 ) on the interval t 1. 1 (8t + 36t 3 ) 1 + 4t 2 + 9t 4 dt = 2 3 (1 + 4t2 + 9t 4 ) 3 2 1 = 2 3 ((14)
More informationAPPM 2350 FINAL EXAM FALL 2017
APPM 25 FINAL EXAM FALL 27. ( points) Determine the absolute maximum and minimum values of the function f(x, y) = 2 6x 4y + 4x 2 + y. Be sure to clearly give both the locations and values of the absolute
More information53. Flux Integrals. Here, R is the region over which the double integral is evaluated.
53. Flux Integrals Let be an orientable surface within 3. An orientable surface, roughly speaking, is one with two distinct sides. At any point on an orientable surface, there exists two normal vectors,
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 informationES.182A Topic 45 Notes Jeremy Orloff
E.8A Topic 45 Notes Jeremy Orloff 45 More surface integrals; divergence theorem Note: Much of these notes are taken directly from the upplementary Notes V0 by Arthur Mattuck. 45. Closed urfaces A closed
More informationMATH 0350 PRACTICE FINAL FALL 2017 SAMUEL S. WATSON. a c. b c.
MATH 35 PRACTICE FINAL FALL 17 SAMUEL S. WATSON Problem 1 Verify that if a and b are nonzero vectors, the vector c = a b + b a bisects the angle between a and b. The cosine of the angle between a and c
More informationAnswers and Solutions to Section 13.7 Homework Problems 1 19 (odd) S. F. Ellermeyer April 23, 2004
Answers and olutions to ection 1.7 Homework Problems 1 19 (odd). F. Ellermeyer April 2, 24 1. The hemisphere and the paraboloid both have the same boundary curve, the circle x 2 y 2 4. Therefore, by tokes
More informationIn general, the formula is S f ds = D f(φ(u, v)) Φ u Φ v da. To compute surface area, we choose f = 1. We compute
alculus III Test 3 ample Problem Answers/olutions 1. Express the area of the surface Φ(u, v) u cosv, u sinv, 2v, with domain u 1, v 2π, as a double integral in u and v. o not evaluate the integral. In
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 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 informationMath 20E Midterm II(ver. a)
Name: olutions tudent ID No.: Discussion ection: Math 20E Midterm IIver. a) Fall 2018 Problem core 1 /24 2 /25 3 /26 4 /25 Total /100 1. 24 Points.) Consider the force field F 5y ı + 3y 2 j. Compute the
More informationJim Lambers MAT 280 Fall Semester Practice Final Exam Solution
Jim Lambers MAT 8 Fall emester 6-7 Practice Final Exam olution. Use Lagrange multipliers to find the point on the circle x + 4 closest to the point (, 5). olution We have f(x, ) (x ) + ( 5), the square
More informationMath 550, Exam 1. 2/10/11.
Math 55, Exam. //. Read problems carefully. Show all work. No notes, calculator, or text. The exam is approximately 5 percent of the total grade. There are points total. Partial credit may be given. Write
More informationPractice Final Solutions
Practice Final Solutions Math 1, Fall 17 Problem 1. Find a parameterization for the given curve, including bounds on the parameter t. Part a) The ellipse in R whose major axis has endpoints, ) and 6, )
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 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 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 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 informationMath 5BI: Problem Set 9 Integral Theorems of Vector Calculus
Math 5BI: Problem et 9 Integral Theorems of Vector Calculus June 2, 2010 A. ivergence and Curl The gradient operator = i + y j + z k operates not only on scalar-valued functions f, yielding the gradient
More informationPractice problems **********************************************************
Practice problems I will not test spherical and cylindrical coordinates explicitly but these two coordinates can be used in the problems when you evaluate triple integrals. 1. Set up the integral without
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 informationMATH2000 Flux integrals and Gauss divergence theorem (solutions)
DEPARTMENT O MATHEMATIC MATH lux integrals and Gauss divergence theorem (solutions ( The hemisphere can be represented as We have by direct calculation in terms of spherical coordinates. = {(r, θ, φ r,
More informationName (please print) π cos(θ) + sin(θ)dθ
Mathematics 2443-3 Final Eamination Form B December 2, 27 Instructions: Give brief, clear answers. I. Evaluate by changing to polar coordinates: 2 + y 2 3 and above the -ais. + y d 23 3 )/3. π 3 Name please
More informationLine and Surface Integrals. Stokes and Divergence Theorems
Math Methods 1 Lia Vas Line and urface Integrals. tokes and Divergence Theorems Review of urves. Intuitively, we think of a curve as a path traced by a moving particle in space. Thus, a curve is a function
More informationName (please print) π cos(θ) + sin(θ)dθ
Mathematics 2443-3 Final Eamination Form A December 2, 27 Instructions: Give brief, clear answers. I. Evaluate by changing to polar coordinates: 2 + y 2 2 and above the -ais. + y d 2(2 2 )/3. π 2 (r cos(θ)
More information(You may need to make a sin / cos-type trigonometric substitution.) Solution.
MTHE 7 Problem Set Solutions. As a reminder, a torus with radii a and b is the surface of revolution of the circle (x b) + z = a in the xz-plane about the z-axis (a and b are positive real numbers, with
More informationMATH 52 FINAL EXAM SOLUTIONS
MAH 5 FINAL EXAM OLUION. (a) ketch the region R of integration in the following double integral. x xe y5 dy dx R = {(x, y) x, x y }. (b) Express the region R as an x-simple region. R = {(x, y) y, x y }
More information( ) = x( u, v) i + y( u, v) j + z( u, v) k
Math 8 ection 16.6 urface Integrals The relationship between surface integrals and surface area is much the same as the relationship between line integrals and arc length. uppose f is a function of three
More information14.7 The Divergence Theorem
14.7 The Divergence Theorem The divergence of a vector field is a derivative of a sort that measures the rate of flow per unit of volume at a point. A field where such flow doesn't occur is called 'divergence
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 informationStokes s Theorem 17.2
Stokes s Theorem 17.2 6 December 213 Stokes s Theorem is the generalization of Green s Theorem to surfaces not just flat surfaces (regions in R 2 ). Relate a double integral over a surface with a line
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 informationSome Important Concepts and Theorems of Vector Calculus
ome Important oncepts and Theorems of Vector alculus 1. The Directional Derivative. If f(x) is a scalar-valued function of, say x, y, and z, andu is a direction (i.e. a unit vector), then the Directional
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 informationMATH 2203 Final Exam Solutions December 14, 2005 S. F. Ellermeyer Name
MATH 223 Final Exam Solutions ecember 14, 25 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation. In
More informationMath 2E Selected Problems for the Final Aaron Chen Spring 2016
Math 2E elected Problems for the Final Aaron Chen pring 216 These are the problems out of the textbook that I listed as more theoretical. Here s also some study tips: 1) Make sure you know the definitions
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 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 informationWORKSHEET #13 MATH 1260 FALL 2014
WORKSHEET #3 MATH 26 FALL 24 NOT DUE. Short answer: (a) Find the equation of the tangent plane to z = x 2 + y 2 at the point,, 2. z x (, ) = 2x = 2, z y (, ) = 2y = 2. So then the tangent plane equation
More informationIdeas from Vector Calculus Kurt Bryan
Ideas from Vector Calculus Kurt Bryan Most of the facts I state below are for functions of two or three variables, but with noted exceptions all are true for functions of n variables..1 Tangent Line Approximation
More informationMath 32B Discussion Session Week 10 Notes March 14 and March 16, 2017
Math 3B iscussion ession Week 1 Notes March 14 and March 16, 17 We ll use this week to review for the final exam. For the most part this will be driven by your questions, and I ve included a practice final
More informationMath 31CH - Spring Final Exam
Math 3H - Spring 24 - Final Exam Problem. The parabolic cylinder y = x 2 (aligned along the z-axis) is cut by the planes y =, z = and z = y. Find the volume of the solid thus obtained. Solution:We calculate
More informationx + ye z2 + ze y2, y + xe z2 + ze x2, z and where T is the
1.(8pts) Find F ds where F = x + ye z + ze y, y + xe z + ze x, z and where T is the T surface in the pictures. (The two pictures are two views of the same surface.) The boundary of T is the unit circle
More informationPractice problems ********************************************************** 1. Divergence, curl
Practice problems 1. Set up the integral without evaluation. The volume inside (x 1) 2 + y 2 + z 2 = 1, below z = 3r but above z = r. This problem is very tricky in cylindrical or Cartesian since we must
More information2. Below are four algebraic vector fields and four sketches of vector fields. Match them.
Math 511: alc III - Practice Eam 3 1. State the meaning or definitions of the following terms: a) vector field, conservative vector field, potential function of a vector field, volume, length of a curve,
More informationFinal Review Worksheet
Score: Name: Final Review Worksheet Math 2110Q Fall 2014 Professor Hohn Answers (in no particular order): f(x, y) = e y + xe xy + C; 2; 3; e y cos z, e z cos x, e x cos y, e x sin y e y sin z e z sin x;
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 informationMATH 255 Applied Honors Calculus III Winter Homework 11. Due: Monday, April 18, 2011
MATH 255 Applied Honors Calculus III Winter 211 Homework 11 ue: Monday, April 18, 211 ection 17.7, pg. 1155: 5, 13, 19, 24. ection 17.8, pg. 1161: 3, 7, 13, 17 ection 17.9, pg. 1168: 3, 7, 19, 25. 17.7
More informationMATH 280 Multivariate Calculus Fall Integrating a vector field over a surface
MATH 280 Multivariate Calculus Fall 2011 Definition Integrating a vector field over a surface We are given a vector field F in space and an oriented surface in the domain of F as shown in the figure below
More information( ) ( ) ( ) ( ) Calculus III - Problem Drill 24: Stokes and Divergence Theorem
alculus III - Problem Drill 4: tokes and Divergence Theorem Question No. 1 of 1 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as needed () Pick the 1. Use
More informationSolutions to the Final Exam, Math 53, Summer 2012
olutions to the Final Exam, Math 5, ummer. (a) ( points) Let be the boundary of the region enclosedby the parabola y = x and the line y = with counterclockwise orientation. alculate (y + e x )dx + xdy.
More information1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r (t) = 3 cos t, 0, 3 sin t, r ( 3π
1. a) 5 points) Find the unit tangent and unit normal vectors T and N to the curve at the point P 3, 3π, r t) 3 cos t, 4t, 3 sin t 3 ). b) 5 points) Find curvature of the curve at the point P. olution:
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 informationMATHEMATICS 317 December 2010 Final Exam Solutions
MATHEMATI 317 December 1 Final Eam olutions 1. Let r(t) = ( 3 cos t, 3 sin t, 4t ) be the position vector of a particle as a function of time t. (a) Find the velocity of the particle as a function of time
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 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 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 informationMcGill University April 20, Advanced Calculus for Engineers
McGill University April 0, 016 Faculty of Science Final examination Advanced Calculus for Engineers Math 64 April 0, 016 Time: PM-5PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer Student
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 information