Mathematics (Course B) Lent Term 2005 Examples Sheet 2
|
|
- Amelia Tyler
- 6 years ago
- Views:
Transcription
1 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 there is a different examples sheet for course A Gradient, Divergence and Curl 1. For the function f(x, y, z) = ln(x 2 + y 2 ) + z, find f. Consider a cylinder of radius 5 whose axis is along the z-axis. (i) What is the rate of change of f(x, y, z) in the direction normal to the cylinder at the point (3, 4, 4)? (ii) What is the rate of change of f(x, y, z) in the direction m = i + 2j at the same point, where i, j are unit vectors parallel to the x, y axes respectively. 2. Find the normal to the surface xz + z 2 xy 2 = 5 at the point (1,1,2). Deduce the equation of the tangent plane at this point. 3. Obtain the equation of the plane that is tangent to the surface z = 3x 2 y sin(πx/2) at the point x = y = 1. Take North to be the direction (0, 1, 0) and East to be the direction (1, 0, 0). In which direction will a marble roll if placed on the surface at x = 1, y = 1 2? 4. A vector field F(x) has components (x 3 + 3y + z 2, y 3, x 2 + y 2 + 3z 2 ). Evaluate (i) the divergence of F(x) and (ii) the curl of F(x). 5. Let a and b be fixed vectors. Show that (a.x) = a. The following are vector functions of x: x; a(x.b); a x; x/r 3, where r = x. Evaluate the divergence and the curl of each of these functions. Line integrals 6. Evaluate Γ {P (x, y)dx + Q(x, y)dy} where P = x 2 y, Q = y 2 x and Γ is the closed curve consisting of the semi-circle x 2 +y 2 = a 2 (y > 0), and the segment ( a, a) of the x-axis, described anti-clockwise. Verify that this is equal to { Q x P } dxdy y where D is the plane surface enclosed by Γ. plane.) D (This illustrates Stokes s theorem in the
2 7. Determine a function f(y) such that [f(y) dx + x cos y dy] = 0 for all closed contours Γ in the (x, y) plane. [Hint: when is the integrand an exact differential?] Γ 8. The work done by a force F acting on a particle which moves along a curve C is defined as the line integral W = F dx. C (i) When F = c v, where c is a constant vector, and v = dx is the velocity of the dt particle, show that the work done is zero. (ii) A particle moves along the helical path given by x = cos t, y = sin t, z = t. Calculate the work W done in the time interval 0 t π by each of the forces F given by (a) yi xj k, (b) xi + yj where i, j, k are unit vectors parallel to the x, y, z axes respectively. 9. Write down a condition obeyed by a conservative vector field V = {P (x, y), Q(x, y)}. Do the following choices for P, Q yield conservative fields? (i) P = x 2 y + y, Q = xy 2 + x ; (ii) P = ye xy + 2x + y, Q = xe xy + x. In the case that V is conservative, find a function f(x, y) such that V = f. Consider the following curves, each joining (0, 0) to (1, 1): (a) C 1 : x = t, y = t (0 t 1) (b) C 2 : x = 0, y = t (0 t 1); x = t, y = 1 (0 t 1). Evaluate the integrals P dx + Qdy and P dx + Qdy for each of the fields (i) and (ii) C 1 C 2 above. 10. Consider the vector field V = (4x 3 z + 2x, z 2 2y, x 4 + 2yz). Evaluate the line integral c V ds along (i) the sequence of straight-line paths joining (0, 0, 0) to (0, 0, 1) to (0, 1, 1) to (1, 1, 1). (ii) the straight line joining (0, 0, 0) to (1, 1, 1), given parametrically by x = y = z = t (0 t 1). Show that V is conservative by finding a function f(x, y, z) such that V = f.
3 Surface integrals 11. Let E = ( ye 2t, xe 2t, 0) and B = (0, 0, e 2t ). Evaluate S B ds, C E dx, where S is the surface of the circular disc x 2 + y 2 < 1, z = 0 and C is the curve bounding S. Show that dx = CE d B ds. dt S 12. A cube is defined by 0 x 1, 0 y 1, 0 z 1. Evaluate the surface integral F nds over the surface of the cube where F = (x 2 +ay 2, 3xy, 6z) and a is a constant. [n is a unit vector in the direction of the outward normal from the volume across a surface element ds; e.g. on the x = 0 face, ds = dydz and n = ( 1, 0, 0).] Evaluate also fdxdydz over the volume of the same cube, where f = bx + 6 and b is a constant. For what values of a and b do these integrals have the same value? (How is this result related to Gauss s law in electricity?) 13. Let u be the vector field u = Qx/4πɛ 0 r 3 in three dimensions, where x is the position vector and r = x (Q, ɛ 0 constants). Show that S u nds = Q/ɛ 0, where S is a sphere of radius a centred on the origin, and n is a unit normal vector pointing radially outward from the sphere. [This is Gauss s law for a point charge.] 14. (i) Find the values of a and b in Question 12 by using the divergence theorem and imposing the requirement that curl F = 0. (ii) For E and B as in Question 11 show show that curl E = B t and apply Stokes s theorem to deduce the equality of the integrals.
4 15. State the divergence theorem. Let F(r) = (x 3 + 3y + z 2, y 3, x 2 + y 2 + 3z 2 ) and let S be the (open) surface 1 z = x 2 + y 2, 0 z 1. Evaluate F nds, where n is the unit normal on S having a positive component in the S z-direction. [Hint: construct a closed surface including S and use the divergence theorem.] 16. State Stokes s theorem, and verify it for the hemispherical surface r = 1, z 0, and the vector field A(x) = (y, x, z). Fourier series 17. Show that the functions 1, x, 1 2 (3x2 1), 1 2 (5x3 3x) are orthogonal on the interval [ 1, 1]. 18. Show that a sin nx sin mx dx = 0 0 if m and n are integers with m n and a = kπ, where k is an integer (i.e. a is an integer multiple of half wavelengths of the fundamental mode, sin x). 19. Prove that for n m (where n and m are integers) T T Find the value of the integral when n = m. sin(mπθ/t ) sin(nπθ/t )dθ = Write down the Fourier series on ( π, π) (with period 2π) for (i) sin 2θ and (ii) cos 2 θ. Obtain the Fourier series for sin 3 θ. 21. Find the Fourier series for the function which equals x when l x l and is periodic with period 2l. If this series is integrated, what will the resulting series sum to? If the series is differentiated, what will the resulting series sum to (if anything)? Give sketches. * Is the rate of convergence of each series roughly what you might expect from considering the smoothness properties of the function? 22. Find the Fourier series with period 2π which converges to e x for π < x < π. To what does it converge when x = π and x = π? * Show that any function f(x) can be written (in terms of f(x) and f( x)) as the sum of an odd function and an even function. Deduce from the first part of the question the Fourier series for cosh x and sinh x with the same range and periodicity.
5 23. Let f(x) = b n sin nx and let g(x) = B m sin mx. Show that m=1 π π f(x)g(x)dx = π b n B n. Without detailed calculation, give the corresponding result when f(x) = a (a n cos nx + b n sin nx) = g(x). 24. Let f(x) = a (a n cos nx + b n sin nx). What can be said about the coefficients a n and b n if: (i) f(x) = f( x) (ii) f(x) = f( x) (iii) f(x) = f(π x) (iv) f(x) = f(π x) (v) f(x) = f(π/2 + x) (vi) f(x) = f(π/2 x) (vii) f(x) = f(2x) (viii) f(x) = f( x) = f(π/2 x). 25. Assume that x 2 = n= c n e inx for π x π, for some complex coefficients c n. By multiplying both sides by e imx and integrating, find the coefficients c n. Write down the (real) Fourier series for x 2 on ( π, π) with period 2π. 26. Let f(x) = x for 0 x < π. Sketch: (i) the odd function; and (ii) the even function that are periodic of period 2π and equal to f(x) for 0 x < π. Show that the function f(x) can be represented for 0 x < π either as or as f(x) = f(x) = 2 π 2( 1) r+1 sin rx r r=1 [ π 2 4 k=0 ] 2 cos(2k + 1)x (2k + 1) 2.
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.
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 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 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 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 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 informationSolutions to old Exam 3 problems
Solutions to old Exam 3 problems Hi students! I am putting this version of my review for the Final exam review here on the web site, place and time to be announced. Enjoy!! Best, Bill Meeks PS. There are
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 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 informationPractice Problems for the Final Exam
Math 114 Spring 2017 Practice Problems for the Final Exam 1. The planes 3x + 2y + z = 6 and x + y = 2 intersect in a line l. Find the distance from the origin to l. (Answer: 24 3 ) 2. Find the area of
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 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 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 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 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 information(b) Find the range of h(x, y) (5) Use the definition of continuity to explain whether or not the function f(x, y) is continuous at (0, 0)
eview Exam Math 43 Name Id ead each question carefully. Avoid simple mistakes. Put a box around the final answer to a question (use the back of the page if necessary). For full credit you must show your
More information16.2. Line Integrals
16. Line Integrals Review of line integrals: Work integral Rules: Fdr F d r = Mdx Ndy Pdz FT r'( t) ds r t since d '(s) and hence d ds '( ) r T r r ds T = Fr '( t) dt since r r'( ) dr d dt t dt dt does
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 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 information29.3. Integral Vector Theorems. Introduction. Prerequisites. Learning Outcomes
Integral ector Theorems 9. Introduction arious theorems exist relating integrals involving vectors. Those involving line, surface and volume integrals are introduced here. They are the multivariable calculus
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 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 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 informationCreated by T. Madas LINE INTEGRALS. Created by T. Madas
LINE INTEGRALS LINE INTEGRALS IN 2 DIMENSIONAL CARTESIAN COORDINATES Question 1 Evaluate the integral ( x + 2y) dx, C where C is the path along the curve with equation y 2 = x + 1, from ( ) 0,1 to ( )
More informationworked out from first principles by parameterizing the path, etc. If however C is a A path C is a simple closed path if and only if the starting point
III.c Green s Theorem As mentioned repeatedly, if F is not a gradient field then F dr must be worked out from first principles by parameterizing the path, etc. If however is a simple closed path in the
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2013 14 CALCULUS AND MULTIVARIABLE CALCULUS MTHA4005Y Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.
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 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 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 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 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 Exam IV - Fall 2011
Math 233 - Exam IV - Fall 2011 December 15, 2011 - Renato Feres NAME: STUDENT ID NUMBER: General instructions: This exam has 16 questions, each worth the same amount. Check that no pages are missing and
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 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 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 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 informationAttempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2017 18 CALCULUS AND MULTIVARIABLE CALCULUS MTHA4005Y Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.
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 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 informationa Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).
Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates
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 information1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is
1. The value of the double integral (a) 15 26 (b) 15 8 (c) 75 (d) 105 26 5 4 0 1 1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order
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 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 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 informationMath Final Exam
Math 221 - Final Exam University of Utah Summer 27 Name: s 1. (1 points) For the vectors: Calculate: (a) (2 points) a + b a = 3i + 2j 2k and b = i + 2j 4k. a + b = ( 3 + ( 1))i + (2 + 2)j + ( 2 + ( 4))k
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 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 informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
NAURAL SCIENCES RIPOS Part IA Wednesday 5 June 2005 9 to 2 MAHEMAICS (2) Before you begin read these instructions carefully: You may submit answers to no more than six questions. All questions carry the
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 informationVector Fields and Line Integrals The Fundamental Theorem for Line Integrals
Math 280 Calculus III Chapter 16 Sections: 16.1, 16.2 16.3 16.4 16.5 Topics: Vector Fields and Line Integrals The Fundamental Theorem for Line Integrals Green s Theorem Curl and Divergence Section 16.1
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 informationPeter Alfeld Math , Fall 2005
WeBWorK assignment due 9/2/05 at :59 PM..( pt) Consider the parametric equation x = 2(cosθ + θsinθ) y = 2(sinθ θcosθ) What is the length of the curve for θ = 0 to θ = 7 6 π? 2.( pt) Let a = (-2 4 2) and
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 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 informationf dr. (6.1) f(x i, y i, z i ) r i. (6.2) N i=1
hapter 6 Integrals In this chapter we will look at integrals in more detail. We will look at integrals along a curve, and multi-dimensional integrals in 2 or more dimensions. In physics we use these integrals
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 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 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 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 informationMathematical Analysis II, 2018/19 First semester
Mathematical Analysis II, 208/9 First semester Yoh Tanimoto Dipartimento di Matematica, Università di Roma Tor Vergata Via della Ricerca Scientifica, I-0033 Roma, Italy email: hoyt@mat.uniroma2.it We basically
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 informationArnie Pizer Rochester Problem Library Fall 2005 WeBWorK assignment VectorCalculus1 due 05/03/2008 at 02:00am EDT.
Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment Vectoralculus due 05/03/008 at 0:00am EDT.. ( pt) rochesterlibrary/setvectoralculus/ur V.pg onsider the transformation T : x = 35 35 37u
More informationMATH 317 Fall 2016 Assignment 5
MATH 37 Fall 26 Assignment 5 6.3, 6.4. ( 6.3) etermine whether F(x, y) e x sin y îı + e x cos y ĵj is a conservative vector field. If it is, find a function f such that F f. enote F (P, Q). We have Q x
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 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 informationLecture 5 - Fundamental Theorem for Line Integrals and Green s Theorem
Lecture 5 - Fundamental Theorem for Line Integrals and Green s Theorem Math 392, section C September 14, 2016 392, section C Lect 5 September 14, 2016 1 / 22 Last Time: Fundamental Theorem for Line Integrals:
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 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 informationIntegration - Past Edexcel Exam Questions
Integration - Past Edexcel Exam Questions 1. (a) Given that y = 5x 2 + 7x + 3, find i. - ii. - (b) ( 1 + 3 ) x 1 x dx. [4] 2. Question 2b - January 2005 2. The gradient of the curve C is given by The point
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-B2a-22 Ver: August 28, 2010 Ver 1.6: Martin Adams, Sep 2009 Ver 1.5: Martin Adams, August 2008 Ver
More informationCITY UNIVERSITY LONDON. BEng (Hons) in Electrical and Electronic Engineering PART 2 EXAMINATION. ENGINEERING MATHEMATICS 2 (resit) EX2003
No: CITY UNIVERSITY LONDON BEng (Hons) in Electrical and Electronic Engineering PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2003 Date: August 2004 Time: 3 hours Attempt Five out of EIGHT questions
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 information4. Line Integrals in the Plane
4. Line Integrals in the Plane 4A. Plane Vector Fields 4A- a) All vectors in the field are identical; continuously differentiable everywhere. b) The vector at P has its tail at P and head at the origin;
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 information1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l.
. If the line l has symmetric equations MA 6 PRACTICE PROBLEMS x = y = z+ 7, find a vector equation for the line l that contains the point (,, ) and is parallel to l. r = ( + t) i t j + ( + 7t) k B. r
More informationChapter 7: Techniques of Integration
Chapter 7: Techniques of Integration MATH 206-01: Calculus II Department of Mathematics University of Louisville last corrected September 14, 2013 1 / 43 Chapter 7: Techniques of Integration 7.1. Integration
More informationName: Instructor: Lecture time: TA: Section time:
Math 222 Final May 11, 29 Name: Instructor: Lecture time: TA: Section time: INSTRUCTIONS READ THIS NOW This test has 1 problems on 16 pages worth a total of 2 points. Look over your test package right
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 informationDirectional Derivative and the Gradient Operator
Chapter 4 Directional Derivative and the Gradient Operator The equation z = f(x, y) defines a surface in 3 dimensions. We can write this as z f(x, y) = 0, or g(x, y, z) = 0, where g(x, y, z) = z f(x, y).
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 informationon an open connected region D, then F is conservative on D. (c) If curl F=curl G on R 3, then C F dr = C G dr for all closed path C.
. (5%) Determine the statement is true ( ) or false ( ). 微甲 -4 班期末考解答和評分標準 (a) If f(x, y) is continuous on the rectangle R = {(x, y) a x b, c y d} except for finitely many points, then f(x, y) is integrable
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 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 informationPage Points Score Total: 210. No more than 200 points may be earned on the exam.
Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Points Score 3 18 4 18 5 18 6 18 7 18 8 18 9 18 10 21 11 21 12 21 13 21 Total: 210 No more than 200
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 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 informationLent 2019 VECTOR CALCULUS EXAMPLE SHEET 1 G. Taylor
Lent 29 ECTOR CALCULU EXAMPLE HEET G. Taylor. (a) The curve defined parametrically by x(t) = (a cos 3 t, a sin 3 t) with t 2π is called an astroid. ketch it, and find its length. (b) The curve defined
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 information(b) Find the interval of convergence of the series whose n th term is ( 1) n (x+2)
Code No: R5112 Set No. 1 I B.Tech Supplimentary Examinations, Aug/Sep 27 MATHEMATICS-I ( Common to Civil Engineering, Electrical & Electronic Engineering, Mechanical Engineering, Electronics & Communication
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 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 informationTom Robbins WW Prob Lib1 Math , Fall 2001
Tom Robbins WW Prob Lib Math 220-2, Fall 200 WeBWorK assignment due 9/7/0 at 6:00 AM..( pt) A child walks due east on the deck of a ship at 3 miles per hour. The ship is moving north at a speed of 7 miles
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 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 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 informationQuestions Q1. The function f is defined by. (a) Show that (5) The function g is defined by. (b) Differentiate g(x) to show that g '(x) = (3)
Questions Q1. The function f is defined by (a) Show that The function g is defined by (b) Differentiate g(x) to show that g '(x) = (c) Find the exact values of x for which g '(x) = 1 (Total 12 marks) Q2.
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 informationTangent Plane. Linear Approximation. The Gradient
Calculus 3 Lia Vas Tangent Plane. Linear Approximation. The Gradient The tangent plane. Let z = f(x, y) be a function of two variables with continuous partial derivatives. Recall that the vectors 1, 0,
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 informationMLC Practice Final Exam
Name: Section: Recitation/Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 13. Show all your work on the standard
More information