Notes 19 Gradient and Laplacian
|
|
- Christina Lamb
- 5 years ago
- Views:
Transcription
1 ECE 3318 Applied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of ECE Notes 19 Gradient and Laplacian 1
2 Gradient Φ ( x, y, z) =scalar function Φ Φ Φ grad Φ xˆ + yˆ + zˆ x y z We can write this as grad Φ= xˆ + yˆ + zˆ Φ x y z Hence grad Φ = Φ 2
3 Directional Derivative Property r We look at how a function changes from one point to a nearby point. d ( xyz,, ) dr (,, ) r + dr x + dx y + dy z + dz dl = dr Recall: ( ) ( ) dφ=φ r + dr Φ r Φ Φ Φ = dx + dy + dz x y z (from calculus) Φ Φ Φ Φ xˆ + yˆ + zˆ x y z ( ) ( ) ( ) dr = xˆ dx + yˆ dy + zˆ dz Hence dφ = Φ dr 3
4 Directional Derivative Property (cont.) r d ( xyz,, ) dr (,, ) r + dr x + dx y + dy z + dz dl = dr Then Use dφ = Φ dr dr = d ( ) dφ = Φ d This gives us the directional derivative: dφ = Φ d Directional derivative 4
5 Physical Interpretation Gradient dφ Φ = Φ d θ Hence dφ = Φ d cosθ dφ Φ=constant = d Direction we march in along path We maximize the directional derivative when we march along in the direction of the gradient (θ = ). The magnitude of the gradient vector gives us the directional derivative when we go in the direction of the gradient. The gradient is perpendicular to a level curve of the function (θ = π / 2). 5
6 Mountain Example Topographic map: Φ(x, y) = height of the landscape at any point. y Φ Φ = + x ( xy, ) xˆ yˆ Φ y Φ x Φ = 1 [m] Φ = [m] Φ = -1 [m] 6
7 Summary of Gradient Formulas Rectangular Φ Φ Φ Φ = xˆ + yˆ + zˆ x y z Cylindrical ˆ 1 Φ = ˆ ρ Φ + φ Φ + ẑ Φ ρ ρ φ z Spherical ˆ1 ˆ 1 Φ = rˆ Φ + θ Φ + φ Φ r r θ rsinθ φ 7
8 Relation Between E and Φ Recall: AB ( ) ( ) V =Φ A Φ B E dr B A Also, from calculus ( ) ( ) Φ A Φ B = dφ = Φ dr A B A B Hence, from the above two results we have B A B E dr = Φ dr = Φ dr A B A 8
9 Relation Between E and Φ (cont.) B A B E dr = Φ dr This must be true for any path. A Assume a small path in the x direction: A x B x B B ( ˆ ) x x x A A x x x B E dr = E x dx = E dx E dx = E x A x B A Similarly, for the second integral: B A ( ) Φ dr Φ x x 9
10 Relation Between E and Φ (cont.) Hence: E x ( ) = Φ x Similarly, using paths in the y and z directions, we have E E y z ( ) ( ) = Φ = Φ y z Hence, we have E = Φ This gives us a new way to find the electric field, by first calculating the potential (illustrated next with examples). Note: The choice of R (the reference point) does not affect E (the gradient of a constant is zero). 1
11 Example Find E from the point charge z x Φ= q 4πε r q [C] [ V] y E Φ ˆ1 Φ ˆ 1 Φ = Φ = rˆ + θ + φ r r θ rsinθ φ = Φ rˆ r q = rˆ 2 4πεr E q = rˆ 2 4πεr [ V/m] 11
12 Line Charge Example y Find E from the line charge ˆ 1 Φ = ˆ ρ Φ + φ Φ + ẑ Φ ρ ρ φ z 2 1 Φ ρ b l Φ= ln 2πε ρ Arbitrary reference point b From previous calculation: Φ = -1 [V] E = E x ρl b = Φ = ˆ ρ ln ρ 2 πε ρ ρl = ˆ ρ ( ln b ln ρ) 2πε ρ = ρ ˆ 2 ρl ˆ ρ 2 πε ρ l ρ πε ρ [V/m] 12
13 Example z Find: E (,, z) (,, z) a R y On the z axis: (,, ) = ˆ (,, ) E z ze z z E = Φ x (,, z) Φ = ρ l [C/m] From previous calculation: 2ε ρ z a + a 2 2 E z Φ = z 13
14 Example (cont.) E z (,, z) so (,, ) ρ dφ z d a = = dz dz 2 2 2ε z + a ρ a 1 Ez z z a z 2ε 2 ( ) ( 2 2) 3/2,, = + ( 2 ) We thus have E z ( z) ρ a ( 2 2 z + a ) [ ],, = V/m 3/2 2ε z 14
15 Vector Identity ( ψ ) = Proof: ( ψ ) xˆ yˆ zˆ x y z ψ ψ ψ x y z = ψ ψ ψ ψ ψ ψ = xˆ yˆ + zˆ yz zy xz zx xy yx = 15
16 Curl Property in Electrostatics (revisited) E = Φ (in statics) E = Φ so = ( ) = Φ ( ) E = 16
17 Equivalent Statements of Path Independence In Statics E = Φ Path Independence E = C E dr = 17
18 Poisson Equation This is a differential equation that the potential satisfies. (This is useful for solving boundary value problems that involve conductors or dielectrics.) Start with the electric Gauss law: D = ρ v ε E = ρ ( ) ( ) v ε Φ = ρ Φ = ( ) v ρ ε v 18
19 Poisson Equation (cont.) Define the Laplacian : Φ Φ Φ ( Φ ) = xˆ + yˆ + zˆ x y z Φ Φ Φ = + + x y z Φ Φ Φ Lap Φ ( Φ ) = + + x y z Poisson s Eq.: Lap Φ= ρv ε 19
20 Poisson Equation (cont.) Del-operator notation for Laplacian: = xˆ + yˆ + zˆ x y z x y z x y z x y z = = xˆ + yˆ + zˆ xˆ + yˆ + zˆ = so x y z Φ= + + Φ = Laplacian operator Hence 2 Lap Φ ( Φ ) = Φ 2
21 Poisson Equation (cont.) Hence, we have 2 Φ= ρ v ε Poisson Equation If ρ v = then 2 Φ= Laplace Equation Note: Φ=Φ ( xyz,, ), ρ = ρ ( xyz,, ) v v 21
22 Poisson Equation (cont.) Siméon Denis Poisson Pierre-Simon Laplace (from Wikipedia) 22
23 Laplacian Rectangular Φ Φ Φ Φ= x y z Cylindrical ρ ρ ρ ρ φ z Φ 1 Φ Φ Φ= ρ Spherical r Φ sinθ Φ Φ Φ= + + r r r r sinθ θ θ r sin θ φ 23
24 Summary of Formulas: Electrostatic Triangle One nice way to summarize all of the equation of electrostatics into one nice visual display is the electrostatic triangle (courtesy of Prof. Donald R. Wilton). 24
25 Electrostatic Triangle ρ v Φ= V ρ v 4πε R dv ( ε E) = ρ v E = V ρ Rˆ v 4πε R S 2 dv D nˆ ds = Q encl 2 ρ Φ= v ε Φ E = Φ E ( ) ( ) Φ r =Φ R E dr r R 25
Notes 3 Review of Vector Calculus
ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018 A ˆ Notes 3 Review of Vector Calculus y ya ˆ y x xa V = x y ˆ x Adapted from notes by Prof. Stuart A. Long 1 Overview Here we present
More informationNotes 24 Image Theory
ECE 3318 Applied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of ECE Notes 24 Image Teory 1 Uniqueness Teorem S ρ v ( yz),, ( given) Given: Φ=ΦB 2 ρv Φ= ε Φ=Φ B on boundary Inside
More informationNotes 4 Electric Field and Voltage
ECE 3318 pplied Electricity and Magnetism Spring 2018 Prof. David R. Jackson Dept. of ECE Notes 4 Electric Field and Voltage Notes prepared by the EM Group University of Houston 1 Electric Field V 0 [
More informationTUTORIAL 7. Discussion of Quiz 2 Solution of Electrostatics part 1
TUTORIAL 7 Discussion of Quiz 2 Solution of Electrostatics part 1 Quiz 2 - Question 1! Postulations of Electrostatics %&''()(*+&,-$'.)/ : % (1)!! E # $$$$$$$$$$ & # (2)!" E # #! Static Electric field is
More informationToday in Physics 217: electric potential
Today in Physics 17: electric potential Finish Friday s discussion of the field from a uniformly-charged sphere, and the gravitational analogue of Gauss Law. Electric potential Example: a field and its
More informationxy 2 e 2z dx dy dz = 8 3 (1 e 4 ) = 2.62 mc. 12 x2 y 3 e 2z 2 m 2 m 2 m Figure P4.1: Cube of Problem 4.1.
Problem 4.1 A cube m on a side is located in the first octant in a Cartesian coordinate system, with one of its corners at the origin. Find the total charge contained in the cube if the charge density
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 informationNotes 18 Faraday s Law
EE 3318 Applied Electricity and Magnetism Spring 2018 Prof. David R. Jackson Dept. of EE Notes 18 Faraday s Law 1 Example (cont.) Find curl of E from a static point charge q y E q = rˆ 2 4πε0r x ( E sinθ
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 informationCreated by T. Madas VECTOR OPERATORS. Created by T. Madas
VECTOR OPERATORS GRADIENT gradϕ ϕ Question 1 A surface S is given by the Cartesian equation x 2 2 + y = 25. a) Draw a sketch of S, and describe it geometrically. b) Determine an equation of the tangent
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 10
ECE 6345 Spring 215 Prof. David R. Jackson ECE Dept. Notes 1 1 Overview In this set of notes we derive the far-field pattern of a circular patch operating in the dominant TM 11 mode. We use the magnetic
More informationENERGY IN ELECTROSTATICS
ENERGY IN ELECTROSTATICS We now turn to the question of energy in electrostatics. The first question to consider is whether or not the force is conservative. You will recall from last semester that a conservative
More informationFundamentals of Applied Electromagnetics. Chapter 2 - Vector Analysis
Fundamentals of pplied Electromagnetics Chapter - Vector nalsis Chapter Objectives Operations of vector algebra Dot product of two vectors Differential functions in vector calculus Divergence of a vector
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 informationAPPLICATIONS OF GAUSS S LAW
APPLICATIONS OF GAUSS S LAW Although Gauss s Law is always correct it is generally only useful in cases with strong symmetries. The basic problem is that it gives the integral of E rather than E itself.
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 informationElectrodynamics PHY712. Lecture 4 Electrostatic potentials and fields. Reference: Chap. 1 & 2 in J. D. Jackson s textbook.
Electrodynamics PHY712 Lecture 4 Electrostatic potentials and fields Reference: Chap. 1 & 2 in J. D. Jackson s textbook. 1. Complete proof of Green s Theorem 2. Proof of mean value theorem for electrostatic
More information3. Calculating Electrostatic Potential
3. Calculating Electrostatic Potential 3. Laplace s Equation 3. The Method of Images 3.3 Separation of Variables 3.4 Multipole Expansion 3.. Introduction The primary task of electrostatics is to study
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 informationENGI Gradient, Divergence, Curl Page 5.01
ENGI 94 5. - Gradient, Divergence, Curl Page 5. 5. The Gradient Operator A brief review is provided here for the gradient operator in both Cartesian and orthogonal non-cartesian coordinate systems. Sections
More informationECE 6340 Intermediate EM Waves. Fall Prof. David R. Jackson Dept. of ECE. Notes 1
EE 6340 Intermediate EM Waves Fall 2016 Prof. David R. Jackson Dept. of EE Notes 1 1 Maxwell s Equations E D rt 2, V/m, rt, Wb/m T ( ) [ ] ( ) ( ) 2 rt, /m, H ( rt, ) [ A/m] B E = t (Faraday's Law) D H
More informationECE 3318 Applied Electricity and Magnetism. Spring Prof. David R. Jackson ECE Dept. Notes 13
ECE 338 Applied Electicity and Magnetism ping 07 Pof. David R. Jackson ECE Dept. Notes 3 Divegence The Physical Concept Find the flux going outwad though a sphee of adius. x ρ v0 z a y ψ = D nˆ d = D ˆ
More informationOmm Al-Qura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics
LECTURE OUTLINE CHAPTER 3 Vectors in Physics 3-1 Scalars Versus Vectors Scalar a numerical value (number with units). May be positive or negative. Examples: temperature, speed, height, and mass. Vector
More informationECE 3318 Applied Electricity and Magnetism Spring 2018 Homework #7
EE 3318 Applied Electricity and Magnetism Spring 2018 Homework #7 Date assigned: Tuesday, March 6, 2018 Date due: Tuesday, March 20, 2018 Do Probs. 1, 2, and 7-12. (You are welcome to do the other problems
More informationGauss s Law. The first Maxwell Equation A very useful computational technique This is important!
Gauss s Law The first Maxwell quation A very useful computational technique This is important! P05-7 Gauss s Law The Idea The total flux of field lines penetrating any of these surfaces is the same and
More informationPreliminary Examination - Day 1 Thursday, August 10, 2017
UNL - Department of Physics and Astronomy Preliminary Examination - Day Thursday, August, 7 This test covers the topics of Quantum Mechanics (Topic ) and Electrodynamics (Topic ). Each topic has 4 A questions
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 informationVector Integrals. Scott N. Walck. October 13, 2016
Vector Integrals cott N. Walck October 13, 16 Contents 1 A Table of Vector Integrals Applications of the Integrals.1 calar Line Integral.........................1.1 Finding Total Charge of a Line Charge..........1.
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 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 informationElectric Field. Electric field direction Same direction as the force on a positive charge Opposite direction to the force on an electron
Electric Field Electric field Space surrounding an electric charge (an energetic aura) Describes electric force Around a charged particle obeys inverse-square law Force per unit charge Electric Field Electric
More informationis the ith variable and a i is the unit vector associated with the ith variable. h i
. Chapter 10 Vector Calculus Features Used right( ), product( ),./,.*, listúmat( ), mod( ), For...EndFor, norm( ), unitv( ),
More informationGeneral review: - a) Dot Product
General review: - a) Dot Product If θ is the angle between the vectors a and b, then a b = a b cos θ NOTE: Two vectors a and b are orthogonal, if and only if a b = 0. Properties of the Dot Product If 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 informationA Brief Revision of Vector Calculus and Maxwell s Equations
A Brief Revision of Vector Calculus and Maxwell s Equations Debapratim Ghosh Electronic Systems Group Department of Electrical Engineering Indian Institute of Technology Bombay e-mail: dghosh@ee.iitb.ac.in
More informationProblem Solving 1: The Mathematics of 8.02 Part I. Coordinate Systems
Problem Solving 1: The Mathematics of 8.02 Part I. Coordinate Systems In 8.02 we regularly use three different coordinate systems: rectangular (Cartesian), cylindrical and spherical. In order to become
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 informationModern Physics. Unit 6: Hydrogen Atom - Radiation Lecture 6.3: Vector Model of Angular Momentum
Modern Physics Unit 6: Hydrogen Atom - Radiation ecture 6.3: Vector Model of Angular Momentum Ron Reifenberger Professor of Physics Purdue University 1 Summary of Important Points from ast ecture The magnitude
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 information6.013 Recitation 11. Quasistatic Electric and Magnetic Fields in Devices and Circuit Elements
6.013 Recitation 11 Quasistatic Electric and Magnetic Fields in Devices and Circuit Elements A. Introduction The behavior of most electric devices depends on static or slowly varying (quasistatic 1 ) electric
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 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 informationxˆ z ˆ. A second vector is given by B 2xˆ yˆ 2z ˆ.
Directions for all homework submissions Submit your work on plain-white or engineering paper (not lined notebook paper). Write each problem statement above each solution. Report answers using decimals
More informationMath review. Math review
Math review 1 Math review 3 1 series approximations 3 Taylor s Theorem 3 Binomial approximation 3 sin(x), for x in radians and x close to zero 4 cos(x), for x in radians and x close to zero 5 2 some geometry
More informationLecture 10 Divergence, Gauss Law in Differential Form
Lecture 10 Divergence, Gauss Law in Differential Form ections: 3.4, 3.5, 3.6 Homework: ee homework file Properties of the Flux Integral: Recap flux is the net normal flow of the vector field F through
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 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 informationMAT 211 Final Exam. Fall Jennings.
MAT 211 Final Exam. Fall 218. Jennings. Useful formulas polar coordinates spherical coordinates: SHOW YOUR WORK! x = rcos(θ) y = rsin(θ) da = r dr dθ x = ρcos(θ)cos(φ) y = ρsin(θ)cos(φ) z = ρsin(φ) dv
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 informationElectromagnetic Field Theory (EMT)
Electromagnetic Field Theory (EMT) Lecture # 9 1) Coulomb s Law and Field Intensity 2) Electric Fields Due to Continuous Charge Distributions Line Charge Surface Charge Volume Charge Coulomb's Law Coulomb's
More informationDIPOLES III. q const. The voltage produced by such a charge distribution is given by. r r'
DIPOLES III We now consider a particularly important charge configuration a dipole. This consists of two equal but opposite charges separated by a small distance. We define the dipole moment as p lim q
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informatione x3 dx dy. 0 y x 2, 0 x 1.
Problem 1. Evaluate by changing the order of integration y e x3 dx dy. Solution:We change the order of integration over the region y x 1. We find and x e x3 dy dx = y x, x 1. x e x3 dx = 1 x=1 3 ex3 x=
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 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 informationy=1 1 J (a x ) dy dz dx dz 10 4 sin(2)e 2y dy dz sin(2)e 2y
Chapter 5 Odd-Numbered 5.. Given the current density J = 4 [sin(x)e y a x + cos(x)e y a y ]ka/m : a) Find the total current crossing the plane y = in the a y direction in the region
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 informationElectrodynamics and Microwaves 3. Gradient, Curl and Divergence
1 Module 3 Gradient, Divergence and Curl 1. Introduction 2. The operators & 2 3. Gradient 4. Divergence 5. Curl 6. Mathematical expressions for gradient, divergence and curl in different coordinate systems.
More informationNote: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2
Math Prelim II Solutions Spring Note: Each problem is worth points except numbers 5 and 6 which are 5 points. x. Compute x da where is the region in the second quadrant between the + y circles x + y and
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 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 informationECE Spring Prof. David R. Jackson ECE Dept. Notes 16
ECE 6345 Spring 5 Prof. David R. Jackson ECE Dept. Notes 6 Overview In this set of notes we calculate the power radiated into space by the circular patch. This will lead to Q sp of the circular patch.
More informationToday in Physics 122: electrostatics review
Today in Physics 122: electrostatics review David Blaine takes the practical portion of his electrostatics midterm (Gawker). 11 October 2012 Physics 122, Fall 2012 1 Electrostatics As you have probably
More information송석호 ( 물리학과 )
http://optics.hanyang.ac.kr/~shsong 송석호 ( 물리학과 ) Introduction to Electrodynamics, David J. Griffiths Review: 1. Vector analysis 2. Electrostatics 3. Special techniques 4. Electric fields in mater 5. Magnetostatics
More informationIn this section, mathematical description of the motion of fluid elements moving in a flow field is
Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small
More informationGradient, Divergence and Curl in Curvilinear Coordinates
Gradient, Divergence and Curl in Curvilinear Coordinates Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found more convenient to work with other coordinate systems.
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 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 informationChapter 1. Vector Algebra and Vector Space
1. Vector Algebra 1.1. Scalars and vectors Chapter 1. Vector Algebra and Vector Space The simplest kind of physical quantity is one that can be completely specified by its magnitude, a single number, together
More 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 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 informationProblem Solving 1: Line Integrals and Surface Integrals
A. Line Integrals MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics Problem Solving 1: Line Integrals and Surface Integrals The line integral of a scalar function f ( xyz),, along a path C is
More informationINSTITUTE OF AERONAUTICAL ENGINEERING Dundigal, Hyderabad DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING
Course Name Course Code Class Branch INSTITUTE OF AERONAUTICAL ENGINEERING Dundigal, Hyderabad - 00 0 DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING : Electro Magnetic fields : A00 : II B. Tech I
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 informationHere are some solutions to the sample problems assigned for Chapter 6.8 to 6.11.
Lecture 3 Appendi B: Some sample problems from Boas Here are some solutions to the sample problems assigned for Chapter 6.8 to 6.. 6.8: Solution: We want to practice doing closed line integrals of the
More informationEE243 Advanced Electromagnetic Theory Lec #3: Electrostatics (Apps., Form),
EE4 Advanced Electromagnetic Theory Lec #: Electrostatics Apps., Form, Electrostatic Boundary Conditions Energy, Force and Capacitance Electrostatic Boundary Conditions on Φ Image Solutions Eample Green
More informationLecture Notes for MATH6106. March 25, 2010
Lecture Notes for MATH66 March 25, 2 Contents Vectors 4. Points in Space.......................... 4.2 Distance between Points..................... 4.3 Scalars and Vectors........................ 5.4 Vectors
More informationTECHNO INDIA BATANAGAR
TECHNO INDIA BATANAGAR ( DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING) QUESTION BANK- 2018 1.Vector Calculus Assistant Professor 9432183958.mukherjee@tib.edu.in 1. When the operator operates on
More informationMP204 Electricity and Magnetism
MATHEMATICAL PHYSICS SEMESTER 2, REPEAT 2016 2017 MP204 Electricity and Magnetism Prof. S. J. Hands, Dr. M. Haque and Dr. J.-I. Skullerud Time allowed: 1 1 2 hours Answer ALL questions MP204, 2016 2017,
More informationTime-Varying Systems; Maxwell s Equations
Time-Varying Systems; Maxwell s Equations 1. Faraday s law in differential form 2. Scalar and vector potentials; the Lorenz condition 3. Ampere s law with displacement current 4. Maxwell s equations 5.
More informationPreliminary Examination - Day 1 Thursday, May 10, 2018
UNL - Department of Physics and Astronomy Preliminary Examination - Day Thursday, May, 28 This test covers the topics of Classical Mechanics (Topic ) and Electrodynamics (Topic 2). Each topic has 4 A questions
More informationMATHEMATICS AS/M/P1 AS PAPER 1
Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks MATHEMATICS AS PAPER 1 Bronze Set B (Edexcel Version) CM Time allowed: 2 hours Instructions to candidates:
More informationMain Results of Vector Analysis
Main Results of ector Analysis Andreas Wacker Mathematical Physics, Lund University January 5, 26 Repetition: ector Space Consider a d dimensional real vector space with scalar product or inner product
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 informationMathematics for Physical Sciences III
Mathematics for Physical Sciences III Change of lecturer: First 4 weeks: myself again! Remaining 8 weeks: Dr Stephen O Sullivan Continuous Assessment Test Date to be announced (probably Week 7 or 8) -
More informationMath 211, Fall 2014, Carleton College
A. Let v (, 2, ) (1,, ) 1, 2, and w (,, 3) (1,, ) 1,, 3. Then n v w 6, 3, 2 is perpendicular to the plane, with length 7. Thus n/ n 6/7, 3/7, 2/7 is a unit vector perpendicular to the plane. [The negation
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 15
ECE 6341 Spring 216 Prof. David R. Jackson ECE Dept. Notes 15 1 Arbitrary Line Current TM : A (, ) Introduce Fourier Transform: I I + ( k ) jk = I e d x y 1 I = I ( k ) jk e dk 2π 2 Arbitrary Line Current
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 information송석호 ( 물리학과 )
http://optics.hanyang.ac.kr/~shsong 송석호 ( 물리학과 ) Field and Wave Electromagnetics, David K. Cheng Reviews on (Week 1). Vector Analysis 3. tatic Electric Fields (Week ) 4. olution of Electrostatic Problems
More informationWelcome. to Electrostatics
Welcome to Electrostatics Outline 1. Coulomb s Law 2. The Electric Field - Examples 3. Gauss Law - Examples 4. Conductors in Electric Field Coulomb s Law Coulomb s law quantifies the magnitude of the electrostatic
More informationAnswer sheet: Final exam for Math 2339, Dec 10, 2010
Answer sheet: Final exam for Math 9, ec, Problem. Let the surface be z f(x,y) ln(y + cos(πxy) + e ). (a) Find the gradient vector of f f(x,y) y + cos(πxy) + e πy sin(πxy), y πx sin(πxy) (b) Evaluate f(,
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 informationOffshore Hydromechanics Module 1
Offshore Hydromechanics Module 1 Dr. ir. Pepijn de Jong 4. Potential Flows part 2 Introduction Topics of Module 1 Problems of interest Chapter 1 Hydrostatics Chapter 2 Floating stability Chapter 2 Constant
More informationLecture 13: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay. Poisson s and Laplace s Equations
Poisson s and Laplace s Equations Lecture 13: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We will spend some time in looking at the mathematical foundations of electrostatics.
More informationG A P between Mathematics and the Physical Sciences
the G A P between Mathematics and the Physical Sciences Tevian Dray & Corinne Manogue Oregon State University http://www.math.oregonstate.edu/bridge Support Mathematical Association of America Professional
More information( ) ( ) QM A1. The operator ˆR is defined by R ˆ ψ( x) = Re[ ψ( x)] ). Is ˆR a linear operator? Explain. (it returns the real part of ψ ( x) SOLUTION
QM A The operator ˆR is defined by R ˆ ψ( x) = Re[ ψ( x)] (it returns the real part of ψ ( x) ). Is ˆR a linear operator? Explain. SOLUTION ˆR is not linear. It s easy to find a counterexample against
More informationChapter 1. Introduction to Electrostatics
Chapter. Introduction to Electrostatics. Electric charge, Coulomb s Law, and Electric field Electric charge Fundamental and characteristic property of the elementary particles There are two and only two
More informationToday in Physics 217: begin electrostatics
Today in Physics 217: begin electrostatics Fields and potentials, and the Helmholtz theorem The empirical basis of electrostatics Coulomb s Law At right: the classic hand-to-thevan-de-graaf experiment.
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