uv = y2 u/v = x 4 y = 1/x 2 u = 1 y = 2/x 2 u = 2
|
|
- Jewel Ellis
- 5 years ago
- Views:
Transcription
1 Section A.. Variable change, divergence and curl (a) x..4 (b) [3 u = x v = /x } uv = u/v = x 4 } = uv x = (u/v) /4 = x v = = x v = = /x u = = /x u = We need to find the Jacobian: J = x u v x v u = u / 4 u 3/4 v /4 v / + u /4 v / 4 v 5/4 u / = = 8 u /4 v 3/4 + 8 u /4 v 3/4 = = 4 u /4 v 3/4 I = = R R uv dx d = J du dv = x R (u/v) /4 u /4 v 3/4 4 u /4 v 3/4 = 4
2 Some students calculated the inverse Jacobian, and directl substituted the values of u and v as a function of x and to obtain the same result (because it cancels out). Although acceptable, the solution above is the correct one. [5 (c)stokes B dr = B da But B is in x plane, so that B is in z direction: B da = [ B z dxd B z = B x B x = x Γ = x dx d = I = R R A majorit of students did use Stokes to obtain the integral above, and related it to the previousl calculated value of I. Minor issues appeared with sorting out the correct sign. [7
3 . Gradients and divergence (a)divergence u = g = 3z (x + )k + xz 3 i + z 3 j = z 3 (xi + j) + 3z (x + )k In polar coordinates, u = z 3 ρ ê ρ + 3z ρ ê k u = x (xz3 ) + (z3 ) + z (3z (x + )) = = z 3 + z 3 + 6z(x + ) = = 4z 3 + 6z(x + ) Cartesian = 4z 3 + 6zρ Polar (b) Gauss: In polar coordinates, u dv = V = u da = 4z 3 dz V u dv [4z 3 + 6zρ πρ dρ dz πρ dρ + 6z dz Both integrals in z are odd in z, and so integrate to zero: u da = πρ 3 dρ [5 Attempts to integrate the fluxes using the full area can succeed, if care is taken to consider the full area, but that approach produced errors, as some students did not consider the cancelation of top and bottom of the clinder. (c) u da = because u = ( g) = (cf. vector identit ( φ) = for an scalar function φ. Alternativel, one can evaluate u using the definition of curl, to find u =. (d)stokes From Stokes theorem, Γ = u dr = C u da S where S is the surface that spans C. But u =, so Γ =. [4 (e)gradient potential v = g g = g[ g x i + g j + g z k = = x ( g )i + ( g )j + z ( g )k = = (g ) [6 [4 3
4 he scalar potential of v is defined b v = φ (or it could be φ), so that φ = ± g We cannot introduce a vector potential for u because the field is not solenoidal, i.e. u. [6 4
5 3. Wave equation (a). We have for the homogeneous solution p, using the intermediate variable ζ = x ± ct: p x = df ζ dζ x + dg ζ dζ x p x = f + g p t = df dζ p t = cf cg ζ t + dg ζ dζ t f + g = c (f + g ) p x = d f ζ dζ x + d g ζ dζ x p x = f + g p t = d f ζ dζ x + d g ζ dζ t p t = c f + c g QED. (b). Using p = X(x) (t): [4 p x = X p t = X c X = X c X X = We then solve two differential equations to ield: = ω X = A cos kx + B sin kx = C cos ωt + D sin ωt where k = ω/c. (c) Since we are seeking a solution that vanishes at x = a, but not at x =, we choose to retain the cosine term (other combinations will give different intermediate solutions, with the same final solution), and incorporate the constant A into the other constants: [5 p(a, t) = cos(ka)(c cos ωt + D sin ωt) = k n a = (n + )π/ ω n = k n c = (c/a)(n + )(π/) p(x, t) = cos(ω n x/c)(c cos ω n t + D sin ω n t) Check: p(, t) = cos(ω n /a)(c cos ω n t + D sin ω n t) p(a, t) = cos ω n a/c(c cos ωt + D sin ωt) = (d). Now the solution at x = must match the given unstead function p sin Ωt. We write the generic solution as a sum of all possible solutions: [5 p(, t) = p sin ωt = cos(ω n x/a)(c cos ω n t + D sin ω n t) n= 5
6 which still obe the initial boundar conditions. We can multipl both sides of the generic solution at x = them b sines or cosines of ω m t and integrate over a time : p sin Ωt cos ω m t dt = p sin Ωt sin ω m t dt = cos(ω n /a) (C }{{} n cos ω n t cos ω m t + D n sin ω n t cos ω m t) dt n= cos(ω n /a) (C }{{} n cos ω n t sin ω m t + D n sin ω n t sin ω m t) dt n= It can be shown that (from Maths datebook, Fourier decomposition) n= n= cos πpt sin πpt πqt cos dt = δ pq πqt sin dt = δ pq c π so that if we make p = n +, q = m +, a = π, or = 4a/c, this corresponds to the Fourier decomposition of p (t), and C n = c a D n = c a 4a/c 4a/c p (t) cos(n + ) π p (t) sin(n + ) π c a t dt c a t dt [7 (e) For a step function at the origin, the step wave will propagate unchanged at the speed of sound through the tube, and reflect at the end, where there is a pressure node. he solution can be approximated b using the same method in (d), where the step function is represented as the sum of harmonics in the corresponding series solution. [4 6
7 exact solu+on t Fourier representa+on of p (t)=h(,t) 7
8 Section B 4. QR decomposition and Least-Squares Solution (a) Q is an othonormal matrix. hus [ c R d = Q v As R is upper-triangular it osthen straight-forward to solve b substituting. [3 (b) he first column is simpl the normalised version of the first column of A and r = 3 q = 3 Projecting the second column and subtracting ields q = 3 he final column is required to be othogonal to the other two. hus Q = 3 ; R = 3 (c) Solving ields (ignoring last column) [ 3 [ c d = 3 [ 8 [8 Solving ields c = 9/9; d = 4/3 (d) For the squared error to be zero, then the last row of (q ) 3 v =. For a solution [ x z = x + z = [6 hus an point ling on the plane will have zero squared error. [8 8
9 = exp ( s σ / sµ) ) [7 5. Sum of random variables and moment generating functions (a): differentiating the expression ields hus mean is λ. Differentiating again ields hus the variance is g (s) = λ exp( λ( s)) λ exp( λ( s)) σ = λ + λ λ = λ (b): the general form is to convolve the two distributions. In this case one is discrete the other continuous. he form must be p z (Z) = p x (X)p (Z X) X= hus the distribution is continuous [4 (c)(i): he MGF can be written g (s) = p (x) exp( sx)dx ( = exp ) πσ σ (x xµ + µ + sxσ ) dx ( = exp ) πσ σ ((x (µ sσ )) s σ 4 + sσ µ) dx [5 (c)(ii) Differentiating this expression ields g z(s) = (sσ µ λ exp( s)) exp ( s σ / sµ + λ(exp( s) ) ) hus the mean is µ z = g z() = µ + λ as expected. he second differential is g z (s) = (σ + λ exp( s)) exp ( s σ / sµ + λ(exp( s) ) ) +(sσ µ λ exp( s)) exp ( s σ / sµ + λ(exp( s) ) ) equating to zero ields and subtracting the µ z σ + λ + (µ + λ) (µ + λ) = σ + λ hese are the values of adding two independent variables. [4 (d): an error occurs when the magnitude of the noise for an integer is greater than.5. For a single integer this can be written as (using the smmetr) Φ(.5/σ) he exception to this is that zero integer value is alwa correctl recogised when the received signal is negative (it will alwas be the closest). hus the overall expression is P e = p x ()Φ(.5/σ) + p x (X)Φ(.5/σ) = ( p x ())Φ(.5/σ) x= = ( exp( λ))φ(.5/σ) 9 [5
10 6. LU Decomposition and Matrix Sub-Spaces (a) Performing LU decomposition ields z = here z 5 = 5 /5 (b) o find the general solution find the solution - solve 4 L = [8 hus = 4 hen solve Ux = = 4 hus setting x 3 = and x 4 = x = o compute the null-space Hence one axis is And the second basis [ [ x x = [ = [ Hence the general solution is x = + α + α 4
11 (c) he left null-space is perpendicular to the column-space - simpl cross-product two columns 7 5 (d) he solution would change as there would be no left-null-space, and there would onl be a single null-space. [3 [ [4
Math 23b Practice Final Summer 2011
Math 2b Practice Final Summer 211 1. (1 points) Sketch or describe the region of integration for 1 x y and interchange the order to dy dx dz. f(x, y, z) dz dy dx Solution. 1 1 x z z f(x, y, z) dy dx dz
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 informationTopic 3. Integral calculus
Integral calculus Topic 3 Line, surface and volume integrals Fundamental theorems of calculus Fundamental theorems for gradients Fundamental theorems for divergences! Green s theorem Fundamental theorems
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 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 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 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 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 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 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 informationSec. 1.1: Basics of Vectors
Sec. 1.1: Basics of Vectors Notation for Euclidean space R n : all points (x 1, x 2,..., x n ) in n-dimensional space. Examples: 1. R 1 : all points on the real number line. 2. R 2 : all points (x 1, x
More information2.20 Fall 2018 Math Review
2.20 Fall 2018 Math Review September 10, 2018 These notes are to help you through the math used in this class. This is just a refresher, so if you never learned one of these topics you should look more
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 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 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 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 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 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 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 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 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 informationChapter 6: Vector Analysis
Chapter 6: Vector Analysis We use derivatives and various products of vectors in all areas of physics. For example, Newton s 2nd law is F = m d2 r. In electricity dt 2 and magnetism, we need surface and
More informationCONSERVATION OF ENERGY FOR ACONTINUUM
Chapter 6 CONSERVATION OF ENERGY FOR ACONTINUUM Figure 6.1: 6.1 Conservation of Energ In order to define conservation of energ, we will follow a derivation similar to those in previous chapters, using
More informationMath 221 Examination 2 Several Variable Calculus
Math Examination Spring Instructions These problems should be viewed as essa questions. Before making a calculation, ou should explain in words what our strateg is. Please write our solutions on our own
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 informationlim = F F = F x x + F y y + F z
Physics 361 Summary of Results from Lecture Physics 361 Derivatives of Scalar and Vector Fields The gradient of a scalar field f( r) is given by g = f. coordinates f g = ê x x + ê f y y + ê f z z Expressed
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 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 informationFinal Exam May 4, 2016
1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.
More informationMEMORIAL UNIVERSITY OF NEWFOUNDLAND
MEMORIAL UNIVERSITY OF NEWFOUNDLAND DEPARTMENT OF MATHEMATICS AND STATISTICS Volumes Math 11 Winter 17 SOLUTIONS 1. (a) i. The axis of smmetr is a horizontal line, so we integrate with respect to x. The
More informationInstructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.
Exam 3 Math 850-007 Fall 04 Odenthal Name: Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.. Evaluate the iterated integral
More informationMath 214 Spring problem set (a) Consider these two first order equations. (I) dy dx = x + 1 dy
Math 4 Spring 08 problem set. (a) Consider these two first order equations. (I) d d = + d (II) d = Below are four direction fields. Match the differential equations above to their direction fields. Provide
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 information1. For each function, find all of its critical points and then classify each point as a local extremum or saddle point.
Solutions Review for Exam # Math 6. For each function, find all of its critical points and then classify each point as a local extremum or saddle point. a fx, y x + 6xy + y Solution.The gradient of f is
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 informationELECTROMAGNETIC WAVES
Physics 4D ELECTROMAGNETIC WAVE Hans P. Paar 26 January 2006 i Chapter 1 Vector Calculus 1.1 Introduction Vector calculus is a branch of mathematics that allows differentiation and integration of (scalar)
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 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 informationMaxwell s equations for electrostatics
Maxwell s equations for electrostatics October 6, 5 The differential form of Gauss s law Starting from the integral form of Gauss s law, we treat the charge as a continuous distribution, ρ x. Then, letting
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 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 informationStochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno
Stochastic Processes M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno 1 Outline Stochastic (random) processes. Autocorrelation. Crosscorrelation. Spectral density function.
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 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 informationMath 350 Solutions for Final Exam Page 1. Problem 1. (10 points) (a) Compute the line integral. F ds C. z dx + y dy + x dz C
Math 35 Solutions for Final Exam Page Problem. ( points) (a) ompute the line integral F ds for the path c(t) = (t 2, t 3, t) with t and the vector field F (x, y, z) = xi + zj + xk. (b) ompute the line
More informationCHAPTER 7 DIV, GRAD, AND CURL
CHAPTER 7 DIV, GRAD, AND CURL 1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: (1 ϕ = ( ϕ, ϕ,, ϕ x 1 x 2 x n
More informationMath 340 Final Exam December 16, 2006
Math 34 Final Exam December 6, 6. () Suppose A 3 4. a) Find the row-reduced echelon form of A. 3 4 so the row reduced echelon form is b) What is rank(a)? 3 4 4 The rank is two since there are two pivots.
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 informationDepartment of Aerospace Engineering AE602 Mathematics for Aerospace Engineers Assignment No. 6
Department of Aerospace Engineering AE Mathematics for Aerospace Engineers Assignment No.. Find the best least squares solution x to x, x 5. What error E is minimized? heck that the error vector ( x, 5
More informationGetting started: CFD notation
PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =
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 informationClassical Field Theory: Electrostatics-Magnetostatics
Classical Field Theory: Electrostatics-Magnetostatics April 27, 2010 1 1 J.D.Jackson, Classical Electrodynamics, 2nd Edition, Section 1-5 Electrostatics The behavior of an electrostatic field can be described
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 informationENGI 9420 Engineering Analysis Solutions to Additional Exercises
ENGI 940 Engineering Analsis Solutions to Additional Exercises 0 Fall [Partial differential equations; Chapter 8] The function ux (, ) satisfies u u u + = 0, subject to the x x u x,0 = u x, =. Classif
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 informationENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 3 Solutions [Multiple Integration; Lines of Force]
ENGI 44 Advanced Calculus for Engineering Facult of Engineering and Applied Science Problem Set Solutions [Multiple Integration; Lines of Force]. Evaluate D da over the triangular region D that is bounded
More informationElectromagnetism Physics 15b
Electromagnetism Physics 15b Lecture #5 Curl Conductors Purcell 2.13 3.3 What We Did Last Time Defined divergence: Defined the Laplacian: From Gauss s Law: Laplace s equation: F da divf = lim S V 0 V Guass
More informationReview Sheet for the Final
Review Sheet for the Final Math 6-4 4 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the absence
More informationMath 67. Rumbos Fall Solutions to Review Problems for Final Exam. (a) Use the triangle inequality to derive the inequality
Math 67. umbos Fall 8 Solutions to eview Problems for Final Exam. In this problem, u and v denote vectors in n. (a) Use the triangle inequality to derive the inequality Solution: Write v u v u for all
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 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 informationENGI 9420 Lecture Notes 8 - PDEs Page 8.01
ENGI 940 ecture Notes 8 - PDEs Page 8.0 8. Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives
More information1. (16 points) Write but do not evaluate the following integrals:
MATH xam # Solutions. (6 points) Write but do not evaluate the following integrals: (a) (6 points) A clindrical integral to calculate the volume of the solid which lies in the first octant (where x,, and
More informationEELE 3331 Electromagnetic I Chapter 3. Vector Calculus. Islamic University of Gaza Electrical Engineering Department Dr.
EELE 3331 Electromagnetic I Chapter 3 Vector Calculus Islamic University of Gaza Electrical Engineering Department Dr. Talal Skaik 2012 1 Differential Length, Area, and Volume This chapter deals with integration
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 informationNST1A: Mathematics II (Course A) End of Course Summary, Lent 2011
General notes Proofs NT1A: Mathematics II (Course A) End of Course ummar, Lent 011 tuart Dalziel (011) s.dalziel@damtp.cam.ac.uk This course is not about proofs, but rather about using different techniques.
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 information(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.
MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.
More informationElectromagnetism HW 1 math review
Electromagnetism HW math review Problems -5 due Mon 7th Sep, 6- due Mon 4th Sep Exercise. The Levi-Civita symbol, ɛ ijk, also known as the completely antisymmetric rank-3 tensor, has the following properties:
More informationFinal exam (practice 1) UCLA: Math 32B, Spring 2018
Instructor: Noah White Date: Final exam (practice 1) UCLA: Math 32B, Spring 2018 This exam has 7 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions
More informationMET 4301 LECTURE SEP17
------------------------------------------------------------------------------------------------------------------------------------ Objectives: To review essential mathematics for Meteorological Dynamics
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 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 information(TRAVELLING) 1D WAVES. 1. Transversal & Longitudinal Waves
(TRAVELLING) 1D WAVES 1. Transversal & Longitudinal Waves Objectives After studying this chapter you should be able to: Derive 1D wave equation for transversal and longitudinal Relate propagation speed
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 informationMaxwell s Equations and Electromagnetic Waves W13D2
Maxwell s Equations and Electromagnetic Waves W13D2 1 Announcements Week 13 Prepset due online Friday 8:30 am Sunday Tutoring 1-5 pm in 26-152 PS 10 due Week 14 Friday at 9 pm in boxes outside 26-152 2
More informationQuick Recapitulation of Fluid Mechanics
Quick Recapitulation of Fluid Mechanics Amey Joshi 07-Feb-018 1 Equations of ideal fluids onsider a volume element of a fluid of density ρ. If there are no sources or sinks in, the mass in it will change
More information3: Mathematics Review
3: Mathematics Review B. Rouben McMaster University Course EP 4D03/6D03 Nuclear Reactor Analysis (Reactor Physics) 015 Sept.-Dec. 015 September 1 Review of: Table of Contents Co-ordinate systems (Cartesian,
More informationPhysics 505 Fall Homework Assignment #9 Solutions
Physics 55 Fall 25 Textbook problems: Ch. 5: 5.2, 5.22, 5.26 Ch. 6: 6.1 Homework Assignment #9 olutions 5.2 a) tarting from the force equation (5.12) and the fact that a magnetization M inside a volume
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 informationChapter 5. Magnetostatics
Chapter 5. Magnetostatics 5.4 Magnetic Vector Potential 5.1.1 The Vector Potential In electrostatics, E Scalar potential (V) In magnetostatics, B E B V A Vector potential (A) (Note) The name is potential,
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 261 Solutions to Sample Final Exam Problems
Math 61 Solutions to Sample Final Eam Problems 1 Math 61 Solutions to Sample Final Eam Problems 1. Let F i + ( + ) j, and let G ( + ) i + ( ) j, where C 1 is the curve consisting of the circle of radius,
More 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 information1 Curvilinear Coordinates
MATHEMATICA PHYSICS PHYS-2106/3 Course Summary Gabor Kunstatter, University of Winnipeg April 2014 1 Curvilinear Coordinates 1. General curvilinear coordinates 3-D: given or conversely u i = u i (x, y,
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 informationFOCUS ON THEORY. We now consider a general change of variable, where x; y coordinates are related to s; t coordinates by the differentiable functions
FOCUS ON HEOY 753 CHANGE OF VAIABLES IN A MULIPLE INEGAL In the previous sections, we used polar, clindrical, and spherical coordinates to simplif iterated integrals. In this section, we discuss more general
More informationPerhaps the simplest way of modeling two (discrete) random variables is by means of a joint PMF, defined as follows.
Chapter 5 Two Random Variables In a practical engineering problem, there is almost always causal relationship between different events. Some relationships are determined by physical laws, e.g., voltage
More informationECE 6340 Intermediate EM Waves. Fall Prof. David R. Jackson Dept. of ECE. Notes 17
ECE 634 Intermediate EM Waves Fall 16 Prof. David R. Jacson Dept. of ECE Notes 17 1 General Plane Waves General form of plane wave: E( xz,, ) = Eψ ( xz,, ) where ψ ( xz,, ) = e j( xx+ + zz) The wavenumber
More informationAPPM 2350 Final Exam points Monday December 17, 7:30am 10am, 2018
APPM 2 Final Exam 28 points Monday December 7, 7:am am, 28 ON THE FONT OF YOU BLUEBOOK write: () your name, (2) your student ID number, () lecture section/time (4) your instructor s name, and () a grading
More informationRadio Propagation Channels Exercise 2 with solutions. Polarization / Wave Vector
/8 Polarization / Wave Vector Assume the following three magnetic fields of homogeneous, plane waves H (t) H A cos (ωt kz) e x H A sin (ωt kz) e y () H 2 (t) H A cos (ωt kz) e x + H A sin (ωt kz) e y (2)
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 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 informationMathematical Notes for E&M Gradient, Divergence, and Curl
Mathematical Notes for E&M Gradient, Divergence, and Curl In these notes I explain the differential operators gradient, divergence, and curl (also known as rotor), the relations between them, the integral
More informationLecture 2 : Curvilinear Coordinates
Lecture 2 : Curvilinear Coordinates Fu-Jiun Jiang October, 200 I. INTRODUCTION A. Definition and Notations In 3-dimension Euclidean space, a vector V can be written as V = e x V x + e y V y + e z V z with
More information10.2-3: Fourier Series.
10.2-3: Fourier Series. 10.2-3: Fourier Series. O. Costin: Fourier Series, 10.2-3 1 Fourier series are very useful in representing periodic functions. Examples of periodic functions. A function is periodic
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 informationSOLUTION OF POISSON S EQUATION. Contents
SOLUTION OF POISSON S EQUATION CRISTIAN E. GUTIÉRREZ OCTOBER 5, 2013 Contents 1. Differentiation under the integral sign 1 2. The Newtonian potential is C 1 2 3. The Newtonian potential from the 3rd Green
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 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 information