ECE Spring Prof. David R. Jackson ECE Dept.
|
|
|
- Felicity Sullivan
- 5 years ago
- Views:
Transcription
1 ECE 6341 Spring 016 Prof. Dvid R. Jckson ECE Dept. Notes Notes
2 Overview In this set of notes we exmine the Arry Scnning Method (ASM for clculting the field of single source ner n infinite periodic structure.
3 ASM Geometry Consider n infinite D periodic rry of metl ptches excited by single (nonperiodic dipole source. z h Side view ε r ( 0,0 L ( x y h,, d 0 0 x Ptches : L W Unit cell : b 3
4 ASM Anlysis We first consider n infinite D periodic rry of metl ptches excited by n infinite periodic rry of dipole sources. z j( kx0m ky0nb Il = e + mn h ε r ( 0,0 L ( x y h,, d 0 0 Il mn x This is n infinite periodic phsed rry problem. 4
5 ASM Anlysis (cont. We use the following identity: π / π / j( kx 0m jm ( π + jm ( π j( kx 0m e e e e dkx0 jm jm π / π / = = = 0 m 0 Picture for m = 1 π < k< π 0 Hence we cn sy tht π / π / e j k ( m x 0 dk x0 0, m 0 = π, m = 0 x 0 e jk Complex plne 5
6 ASM Anlysis (cont. Denote (,, ;, E xyzk k x x0 y0 = field produced by infinite periodic rry problem with phsing ( k, k x0 y0 Then π π / π / (,, ;, E x y z k k dk x x0 y0 x0 = field produced by single column of dipole sources 6
7 ASM Anlysis (cont. (,, ;, E xyzk k x x0 y0 = field from D rry of phsed dipoles y x 7
8 ASM Anlysis (cont. π π / π / (,, ;, E x y z k k dk x x0 y0 x0 = field from single column of dipoles (phsed in the y direction y x 8
9 ASM Anlysis (cont. Next, we pply the sme procedure to the phsing in the y direction: π / b π / b e j k ( y 0mb dk y0 0, m 0 = π, m = 0 b 9
10 ASM Anlysis (cont. b ( π π/ b π/ π/ b π/ (,, ;, E x y z k k dk dk x x0 y0 x0 y0 y = field from single dipole x 10
11 ASM Anlysis (cont. Conclusion: π/ b π/ b x,, =,, ; x x0, y0 x0 y0 π/ b π/ E x y z E x y z k k dk dk ( π y x 11
12 ASM Anlysis (cont. After doing the method of moments (plese see the Appendix, the result for the infinite phsed rry problem will be in the form x x y xp yq p= q= (,, ; 0, 0 =, ; E xyzk k Ak k z e ( xp yq jk x+ k y Floquet expnsion 1
13 ASM Anlysis (cont. b jk ( xpx+ kyq y E x, y, z A k, k ; z e dk dk ( π π/ b π/ = x xp yq x0 y0 π/ b π/ p= q= Plese see the next slide. b jk ( x0x+ ky0y E x, y, z A k, k ; z e dk dk ( π = x x0 y0 x0 y0 This is the unfolded form (the integrtion limits re infinite. 13
14 ASM Anlysis (cont. Physicl explntion of the pth unfolding (illustrted for the k x0 integrl: k xp = k + x0 π p p = 1 p = 0 p = 1 π π k x0 Fundmentl Brillouin zone 14
15 Appendix In this ppendix we use the method of moments to clculte (,, ; 0, 0 E xyzk k x x y 15
16 Appendix (cont. Assume tht unknown current on the (0,0 ptch in the D rry problem is of the following form: The EFIE is then (, = (, J xy A B xy sx x x π x Bx ( xy, = cos, x< L/, y< W/ L dip Ax E x B x + E x J sx = 0, x < L/, y < W / Note tht the superscript stnds for infinite periodic (i.e., the fields due to the infinite periodic rry of ptch currents. The EFIE is enforced on the (0,0 ptch; it is then utomticlly enforced on ll ptches. 16
17 Appendix (cont. We hve, using Glerkin s method, 0 0 A B ( x, y E B ds + B x, y E J ds = 0 dip x x x x x x sx S S Define Z = B ( x, y E B ds xx x x x S 0 0 (, R = B x y E J ds dip x x sx S We then hve A Z x xx = R 17
18 Appendix (cont. The (0,0 ptch current mplitude is then (, A k k x x0 y0 = x0, y0 (, R k k Z k k xx x0 y0 We lso hve single 1 (, (, xx = xx x y x x y x y Z G k k B k k dk dk ( π ( π 1, (, xx = xx xp yq x xp yq b p= q= Z G k k B k k ( π 18
19 Appendix (cont. For the RHS term we hve 0 0 (, R = B x y E J ds dip x x sx S (, J x y E B ds dip = sx x x S (,, = E B x y h x x 0 0 d This follows from reciprocity for single unit cell together with the periodic SDI method. The field from the periodic rry of ptch bsis functions is ( π 1 x ( x0 0 x = xx xp yq b p= q= ( π E xyzk,, ;, k B G k, k ; z,0 (, B k k e x xp yq ( xp yq jk x+ k y 19
20 Appendix (cont. Hence, we hve ( π 1 x0, y0 =, ;,0, xx xp yq d x xp yq b p= q= ( π R k k G k k h B k k e ( xp 0 yq 0 jk x+ k y where L cos k π W B k k LW k π L kxp xp x xp, yq = sinc yq 0
21 Appendix (cont. We then hve, for the contribution due to the ptches: π 1 E x y z k k A k k G k k z,, ; 0, 0 = 0, 0 (, ;,0 b p= q= ( π xp + yq B x ( kxp, kyq e, ptches x x y x x y xx xp yq jk x k y For the contribution due to the dipoles: ( π, dipoles 1 x (,, ; x0, y0 =, ;, xx xp yq d b p= q= ( π E x y z k k Il G k k z h ( 1 e ( xp yq jk x+ k y 1
22 Appendix (cont. We hve tht 0, 0 =, A k k A k k x x y x xp yq (This is becuse we hve the sme physicl dipole excittion for either set of phsing wvenumbers. We then hve ( π, ptches 1 x ( x0 y0 = x xp yq xx xp yq b p= q= ( π E x, y, z; k, k A k, k G k, k ; z,0 (, B k k e x xp yq ( xp yq jk x+ k y
23 Appendix (cont. We then use, dipoles, ptches,, ;, =,, ;, + (,, ;, E xyzk k E xyzk k E xyzk k x x y x x y x x y so tht ( π ( π 1 x (,, ; x0, y0 =,, ;,0 x xp yq xx xp yq b p= q= 1 ( π ( π xx ( xp yq d Il G k, k ; z, h e E x y z k k A k k G k k z + b p= q= (, B k k e x xp yq ( xp yq jk x+ k y ( xp yq jk x+ k y 3
24 Appendix (cont. We therefore identify ( π 1 xp, yq; =,, ;,0, x xp yq xx xp yq x xp yq ( π ( π A k k z A k k G k k z B k k b 1 + Il G G k k z h b ( π (, ;, xx xx xp yq d 4
25 Appendix (cont. Note: When clculting the field in the originl problem, there is no need to use the ASM to find the fields from the originl (single dipole; we cn lso find this directly using the (non-periodic SDI method. x We then hve dipole (,, = (,, E xyz E xyz x π/ b π/ b π 1 + A x kx0, ky0 G xx ( kxp, kyq; z,0 ( π b π/ b π/ p= q= ( π jk B ( xpx+ kyq y k k e dk dk (, x xp yq x0 y0 5
ECE Spring Prof. David R. Jackson ECE Dept. Notes 32
ECE 6345 Spring 215 Prof. David R. Jackson ECE Dept. Notes 32 1 Overview In this set of notes we extend the spectral-domain method to analyze infinite periodic structures. Two typical examples of infinite
ECE Spring Prof. David R. Jackson ECE Dept. Notes 33
C 6345 Spring 2015 Prof. David R. Jackson C Dept. Notes 33 1 Overview In this set of notes we eamine the FSS problem in more detail, using the periodic spectral-domain Green s function. 2 FSS Geometry
ECE Spring Prof. David R. Jackson ECE Dept. Notes 26
ECE 6345 Spring 05 Prof. David R. Jackson ECE Dept. Notes 6 Overview In this set of notes we use the spectral-domain method to find the mutual impedance between two rectangular patch ennas. z Geometry
ECE 3318 Applied Electricity and Magnetism. Spring Prof. David R. Jackson Dept. of ECE. Notes 31 Inductance
ECE 3318 Applied Electricity nd Mgnetism Spring 018 Prof. Dvid R. Jckson Dept. of ECE Notes 31 nductnce 1 nductnce ˆn S Single turn coil The current produces flux though the loop. Definition of inductnce:
Summary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
Matrix Eigenvalues and Eigenvectors September 13, 2017
Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues
Waveguides Free Space. Modal Excitation. Daniel S. Weile. Department of Electrical and Computer Engineering University of Delaware
Modl Excittion Dniel S. Weile Deprtment of Electricl nd Computer Engineering University of Delwre ELEG 648 Modl Excittion in Crtesin Coordintes Outline 1 Aperture Excittion Current Excittion Outline 1
n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014
SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.
Improper Integrals. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Improper Integrls MATH 2, Clculus II J. Robert Buchnn Deprtment of Mthemtics Spring 28 Definite Integrls Theorem (Fundmentl Theorem of Clculus (Prt I)) If f is continuous on [, b] then b f (x) dx = [F(x)]
Section 7.1 Integration by Substitution
Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find
When a force f(t) is applied to a mass in a system, we recall that Newton s law says that. f(t) = ma = m d dt v,
Impulse Functions In mny ppliction problems, n externl force f(t) is pplied over very short period of time. For exmple, if mss in spring nd dshpot system is struck by hmmer, the ppliction of the force
Kai Sun. University of Michigan, Ann Arbor
Ki Sun University of Michign, Ann Arbor How to see toms in solid? For conductors, we cn utilize scnning tunneling microscope (STM) to see toms (Nobel Prize in Physics in 1986) Limittions: (1) conductors
Jackson 2.7 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson.7 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: Consider potentil problem in the hlf-spce defined by, with Dirichlet boundry conditions on the plne
Improper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
Polynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
Lecture 13 - Linking E, ϕ, and ρ
Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on
Section 17.2 Line Integrals
Section 7. Line Integrls Integrting Vector Fields nd Functions long urve In this section we consider the problem of integrting functions, both sclr nd vector (vector fields) long curve in the plne. We
AMATH 731: Applied Functional Analysis Fall Additional notes on Fréchet derivatives
AMATH 731: Applied Functionl Anlysis Fll 214 Additionl notes on Fréchet derivtives (To ccompny Section 3.1 of the AMATH 731 Course Notes) Let X,Y be normed liner spces. The Fréchet derivtive of n opertor
df dx There is an infinite number of different paths from
Integrl clculus line integrls Feb 7, 18 From clculus, in the cse of single vrible x1 F F x F x f x dx, where f x 1 x df dx Now, consider the cse tht two vribles re t ply. Suppose,, df M x y dx N x y dy
Numerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
Physics 3323, Fall 2016 Problem Set 7 due Oct 14, 2016
Physics 333, Fll 16 Problem Set 7 due Oct 14, 16 Reding: Griffiths 4.1 through 4.4.1 1. Electric dipole An electric dipole with p = p ẑ is locted t the origin nd is sitting in n otherwise uniform electric
Electromagnetism Answers to Problem Set 10 Spring 2006
Electromgnetism 76 Answers to Problem Set 1 Spring 6 1. Jckson Prob. 5.15: Shielded Bifilr Circuit: Two wires crrying oppositely directed currents re surrounded by cylindricl shell of inner rdius, outer
ECE 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
df dt f () b f () a dt
Vector lculus 16.7 tokes Theorem Nme: toke's Theorem is higher dimensionl nlogue to Green's Theorem nd the Fundmentl Theorem of clculus. Why, you sk? Well, let us revisit these theorems. Fundmentl Theorem
Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
Department of Electrical and Computer Engineering, Cornell University. ECE 4070: Physics of Semiconductors and Nanostructures.
Deprtment of Electricl nd Computer Engineering, Cornell University ECE 4070: Physics of Semiconductors nd Nnostructures Spring 2014 Exm 2 ` April 17, 2014 INSTRUCTIONS: Every problem must be done in the
We divide the interval [a, b] into subintervals of equal length x = b a n
![We divide the interval [a, b] into subintervals of equal length x = b a n](/thumbs/80/80533932.jpg "We divide the interval [a, b] into subintervals of equal length x = b a n")
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
Review on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.
Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5. - 5.3) Remrks on the course. Slide Review: Sec. 5.-5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description
x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x " 0 :
Clculus II MAT 146 Integrtion Applictions: Volumes of 3D Solids Our gol is to determine volumes of vrious shpes. Some of the shpes re the result of rotting curve out n xis nd other shpes re simply given
The Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
NUMERICAL INTEGRATION
NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls
Today in Physics 122: work, energy and potential in electrostatics
Tody in Physics 1: work, energy nd potentil in electrosttics Leftovers Perfect conductors Fields from chrges distriuted on perfect conductors Guss s lw for grvity Work nd energy Electrosttic potentil energy,
HOMEWORK SOLUTIONS MATH 1910 Sections 7.9, 8.1 Fall 2016
HOMEWORK SOLUTIONS MATH 9 Sections 7.9, 8. Fll 6 Problem 7.9.33 Show tht for ny constnts M,, nd, the function yt) = )) t ) M + tnh stisfies the logistic eqution: y SOLUTION. Let Then nd Finlly, y = y M
Lecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
Week 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
December 4, U(x) = U 0 cos 4 πx 8
PHZ66: Fll 013 Problem set # 5: Nerly-free-electron nd tight-binding models: Solutions due Wednesdy, 11/13 t the time of the clss Instructor: D L Mslov mslov@physufledu 39-0513 Rm 11 Office hours: TR 3
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
We know that if f is a continuous nonnegative function on the interval [a, b], then b
![We know that if f is a continuous nonnegative function on the interval [a, b], then b](/thumbs/72/66633133.jpg "We know that if f is a continuous nonnegative function on the interval [a, b], then b")
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
Read section 3.3, 3.4 Announcements:
Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f
We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
![We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.](/thumbs/93/114277851.jpg "We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.")
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
Physics 202, Lecture 14
Physics 202, Lecture 14 Tody s Topics Sources of the Mgnetic Field (Ch. 28) Biot-Svrt Lw Ampere s Lw Mgnetism in Mtter Mxwell s Equtions Homework #7: due Tues 3/11 t 11 PM (4th problem optionl) Mgnetic
Convex Sets and Functions
B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line
ECE 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
ENGI 3424 Engineering Mathematics Five Tutorial Examples of Partial Fractions
ENGI 44 Engineering Mthemtics Five Tutoril Exmples o Prtil Frctions 1. Express x in prtil rctions: x 4 x 4 x 4 b x x x x Both denomintors re liner non-repeted ctors. The cover-up rule my be used: 4 4 4
4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
Wave Phenomena Physics 15c
Wve Phenomen Physics 15c Lecture Diffrction (H&L Chpter 11) Wht We Did Lst Time! Studied interference! or more wves overlp " Amplitudes dd up " Intensity = (mplitude) does not dd up! Thin-film interference!
ragsdale (zdr82) HW2 ditmire (58335) 1
rgsdle (zdr82) HW2 ditmire (58335) This print-out should hve 22 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc
5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship
5.4, 6.1, 6.2 Hnout As we ve iscusse, the integrl is in some wy the opposite of tking erivtive. The exct reltionship is given by the Funmentl Theorem of Clculus: The Funmentl Theorem of Clculus: If f is
13.4 Work done by Constant Forces
13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push
(6.5) Length and area in polar coordinates
86 Chpter 6 SLICING TECHNIQUES FURTHER APPLICATIONS Totl mss 6 x ρ(x)dx + x 6 x dx + 9 kg dx + 6 x dx oment bout origin 6 xρ(x)dx x x dx + x + x + ln x ( ) + ln 6 kg m x dx + 6 6 x x dx Centre of mss +
1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
Anonymous Math 361: Homework 5. x i = 1 (1 u i )
Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define
1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous
Lecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics - A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
ECE Spring Prof. David R. Jackson ECE Dept.
ECE 634 Spring 26 Prof. David R. Jacson ECE Dept. Notes Notes 42 43 Sommerfeld Problem In this set of notes we use SDI theory to solve the classical "Sommerfeld problem" of a vertical dipole over an semi-infinite
1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection
CS683: calculating the effective resistances
CS683: clculting the effective resistnces Lecturer: John Hopcroft Note tkers: June Andrews nd Jen-Bptiste Jennin Mrch 7th, 2008 On Ferury 29th we sw tht, given grph in which ech edge is lelled with resistnce
Math 120 Answers for Homework 13
Mth 12 Answers for Homework 13 1. In this problem we will use the fct tht if m f(x M on n intervl [, b] (nd if f is integrble on [, b] then (* m(b f dx M(b. ( The function f(x = 1 + x 3 is n incresing
Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 6 (First moments of an arc) A.J.Hobson
JUST THE MATHS UNIT NUMBER 13.6 INTEGRATION APPLICATIONS 6 (First moments of n rc) by A.J.Hobson 13.6.1 Introduction 13.6. First moment of n rc bout the y-xis 13.6.3 First moment of n rc bout the x-xis
Math 1431 Section 6.1. f x dx, find f. Question 22: If. a. 5 b. π c. π-5 d. 0 e. -5. Question 33: Choose the correct statement given that
Mth 43 Section 6 Question : If f d nd f d, find f 4 d π c π- d e - Question 33: Choose the correct sttement given tht 7 f d 8 nd 7 f d3 7 c d f d3 f d f d f d e None of these Mth 43 Section 6 Are Under
Math 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
AP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
Point Lattices: Bravais Lattices
Physics for Solid Stte Applictions Februry 18, 2004 Lecture 7: Periodic Structures (cont.) Outline Review 2D & 3D Periodic Crystl Structures: Mthemtics X-Ry Diffrction: Observing Reciprocl Spce Point Lttices:
Lecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
1 Functions Defined in Terms of Integrals
November 5, 8 MAT86 Week 3 Justin Ko Functions Defined in Terms of Integrls Integrls llow us to define new functions in terms of the bsic functions introduced in Week. Given continuous function f(), consider
Chapter 4. Lebesgue Integration
4.2. Lebesgue Integrtion 1 Chpter 4. Lebesgue Integrtion Section 4.2. Lebesgue Integrtion Note. Simple functions ply the sme role to Lebesgue integrls s step functions ply to Riemnn integrtion. Definition.
Chapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
HW Solutions # MIT - Prof. Kowalski. Friction, circular dynamics, and Work-Kinetic Energy.
HW Solutions # 5-8.01 MIT - Prof. Kowlski Friction, circulr dynmics, nd Work-Kinetic Energy. 1) 5.80 If the block were to remin t rest reltive to the truck, the friction force would need to cuse n ccelertion
On the Decomposition Method for System of Linear Fredholm Integral Equations of the Second Kind
Applied Mthemticl Sciences, Vol. 2, 28, no. 2, 57-62 On the Decomposition Method for System of Liner Fredholm Integrl Equtions of the Second Kind A. R. Vhidi 1 nd M. Mokhtri Deprtment of Mthemtics, Shhr-e-Rey
Now, given the derivative, can we find the function back? Can we antidifferenitate it?
Fundmentl Theorem of Clculus. Prt I Connection between integrtion nd differentition. Tody we will discuss reltionship between two mjor concepts of Clculus: integrtion nd differentition. We will show tht
Explain shortly the meaning of the following eight words in relation to shells structures.
Delft University of Technology Fculty of Civil Engineering nd Geosciences Structurl Mechnics Section Write your nme nd study number t the top right-hnd of your work. Exm CIE4143 Shell Anlysis Tuesdy 15
Green function and Eigenfunctions
Green function nd Eigenfunctions Let L e regulr Sturm-Liouville opertor on n intervl (, ) together with regulr oundry conditions. We denote y, φ ( n, x ) the eigenvlues nd corresponding normlized eigenfunctions
This final is a three hour open book, open notes exam. Do all four problems.
Physics 55 Fll 27 Finl Exm Solutions This finl is three hour open book, open notes exm. Do ll four problems. [25 pts] 1. A point electric dipole with dipole moment p is locted in vcuum pointing wy from
Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
Integration Techniques
Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u
Overview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
Lecture 4: Piecewise Cubic Interpolation
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics - A Peirce UBC Lecture 4: Piecewise Cubic Interpoltion Compiled 5 September In this lecture we consider piecewise cubic interpoltion
A B= ( ) because from A to B is 3 right, 2 down.
8. Vectors nd vector nottion Questions re trgeted t the grdes indicted Remember: mgnitude mens size. The vector ( ) mens move left nd up. On Resource sheet 8. drw ccurtely nd lbel the following vectors.
Best Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes
Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl
Introduction to Group Theory
Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements
MTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008
MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul
Homework Assignment 9 Solution Set
Homework Assignment 9 Solution Set PHYCS 44 3 Mrch, 4 Problem (Griffiths 77) The mgnitude of the current in the loop is loop = ε induced = Φ B = A B = π = π µ n (µ n) = π µ nk According to Lense s Lw this
Homework Assignment 3 Solution Set
Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.
If deg(num) deg(denom), then we should use long-division of polynomials to rewrite: p(x) = s(x) + r(x) q(x), q(x)
Mth 50 The method of prtil frction decomposition (PFD is used to integrte some rtionl functions of the form p(x, where p/q is in lowest terms nd deg(num < deg(denom. q(x If deg(num deg(denom, then we should
1. Gauss-Jacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. Guss-Jcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
MATH 253 WORKSHEET 24 MORE INTEGRATION IN POLAR COORDINATES. r dr = = 4 = Here we used: (1) The half-angle formula cos 2 θ = 1 2
MATH 53 WORKSHEET MORE INTEGRATION IN POLAR COORDINATES ) Find the volume of the solid lying bove the xy-plne, below the prboloid x + y nd inside the cylinder x ) + y. ) We found lst time the set of points
Note 16. Stokes theorem Differential Geometry, 2005
Note 16. Stokes theorem ifferentil Geometry, 2005 Stokes theorem is the centrl result in the theory of integrtion on mnifolds. It gives the reltion between exterior differentition (see Note 14) nd integrtion
Overview of Calculus
Overview of Clculus June 6, 2016 1 Limits Clculus begins with the notion of limit. In symbols, lim f(x) = L x c In wors, however close you emn tht the function f evlute t x, f(x), to be to the limit L
Physics 220. Exam #1. April 21, 2017
Physics Exm # April, 7 Nme Plese red nd follow these instructions crefully: Red ll problems crefully before ttempting to solve them. Your work must be legible, nd the orgniztion cler. You must show ll