Line Integrals Independence of Path -(9.9) f x
|
|
- Ashlyn Boone
- 5 years ago
- Views:
Transcription
1 . Differentials and Eact Differentials: The differentials of f, and f,,z are: Line Integrals Independence of Path -(9.9) d d, and d d dz. A differential epression P,,zd Q,,zd R,,zdz is said to be an eact differential if there eists a function f,,z such that P,,z d Q,,z d R,,z dz that is f,,z P,,z, f,,z Q,,z, f z,,z R,,z. (Remember eact differential equations in differential equations?!) Eample Let f,,z e z ompute the differential. e z z e z d e z d 3/ z e z dz Eample Determine if the differential d d is eact. Two was to determine if a differential is eact: a. If it is eact, then there eists a function f, such that f, d 4, Such a does not eist since the right side of the equation is a function of and. This contradicts to the assumption of the eistence of f,. Therefore, the differential is not eact. b. If and for some f,, then since both and are polnomials. heck 4, f. Since, f, does eists and the differential is not eact.
2 Note that for f,,z we need to check three equations:.. Path Independence: Let A and B be two points in a space (a plane). A line integral F dr is said to be independent of the path if the value of the integral is the same for ever curve connecting A and B. Eample ompute F dr where F, i, and : from, to, b a straight line; : from, to, b the path: ; 3 : from, to, b the path: rt t 3/, t t t 3/, t t 5 : r t t,t, t ; : r t t, t, t ; 3 : r 3 t t 3/,t 5, t F dr t,t, t t F dr t,t, t t t 3 3 t3/ 3 F dr t 5, t 3/ 5t 4, 3 t / 5t 9 3 t t t3 Observe that F dr are the same for i,,3. Though we cannot conclude from these 3 eamples i that this line integral is independent of the path, we see it is possible that the line integral is independent of the path. A Fundamental Theorem for Line Integrals: Suppose that
3 P,,z d Q,,z d R,,z dz (the differential P,,z d Q,,z d R,,z dz is eact) and is a curve from A to B. Then a. the line integral P,,z d Q,,z d R,,z dz is independent of the path ; and b. P,,zd Q,,zd R,,z dz fb fa Proof: Let rt t, t, zt, a t b be a parametric representation of where A a,a,za and B b,b,zb. Then we know d d dz and P,,z d Q,,z d Now we can rewrite the line integral as: P,,zd Q,,zd R,,zdz P,,z d a b a b R,,z dz. d Q,,z d d dz R,,z dz ft,t,zt a b fb fa Notes: LetF,,z P,,z, Q,,z, R,,z for,,z in a region D which contains. Observe that F dr P,,z d Q,,z d R,,z dz. If the differential P,,z d Q,,z d R,,z dz is eact, then the integral F dr is independent of the path. If F dr is independent of the path, then F dr. If P,,zd Q,,zd R,,zdz is eact, then we know there eists a function f,,z such that P,,z, Q,,z, R,,z, that is f,,z F,,z. So, for a given F,,z, if there eists a scalar function f,,z such that f,,z F,,z, then F dr is path independent and F dr fb fa onservative vector fields: A vector field F,,z is said to be conservative if there is a scalar function f,,z such that f F. f is called a potential function of F and F is called a gradient field. 3
4 Note that if a force field F is conservative then the work done b F from the point A to the point B in space is the same for an path from A to B. If F dr depends on the path, then F is not conservative. A frictional force such as air resistance is not conservative. Two Was to Evaluate a Path Independent Line Integral: Let F dr be independent of path. Find f such that f F and then F dr fb fa. hose a curve from A to B and evaluate F dr. Eample Let F,,z 3,z, andbeacurvefroma,, to B,,7.ompute F dr in two was. a. Fin such that f F. 3, f,,z 3 d 3,z. heck z.,z zd z z, f,,z 3 z z. heck z z, z. f,,z 3 z F dr fb fa f,,7 f,, 7 6 b. Let rt,, t,,5 t, t, 5t.AtB, t, at A, t F dr 3t, t 5t, t,,5 3t 4 t 5t 5 t t 3 4 t 5 t 3 t3 5 t 3 t Test for Path Independence: Let F,,z P,,z, Q,,z, R,,z for,,z in an open simpl connected region D which contains where P, Q and R have continuous first partial derivatives. Then F dr is independent of the path if and onl if curl F,,z. Note that curl F,,z Q, Q, P P, Q, Q P P. implies F dr is independent of the path implies there eists a potential function f,,z such f F, that is 4
5 P, Q, R. Since P, Q and R have continuous first partial derivatives, f has continuous second partial derivatives. So, That is P Q ;,, P ;. Q. Eample Let F z, z zcosz, z cosz. Determine if F is conservative and,, evaluate F dr.,, Determine first if F is a gradient field of f. z, f,,z z d,z z,z z,z z zcosz,,z zcosz,,z zcoszd,,z sinz z, f,,z z sinz z z cosz z z cosz, z, z f,,z z sinz. So, F is conservative and its potential function is f,,z z sinz,, F dr f,, f,,,, 5
Conservative fields and potential functions. (Sect. 16.3) The line integral of a vector field along a curve.
onservative fields and potential functions. (Sect. 16.3) eview: Line integral of a vector field. onservative fields. The line integral of conservative fields. Finding the potential of a conservative field.
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 informationComplex Analysis Topic: Singularities
Complex Analysis Topic: Singularities MA201 Mathematics III Department of Mathematics IIT Guwahati August 2015 Complex Analysis Topic: Singularities 1 / 15 Zeroes of Analytic Functions A point z 0 C is
More informationMAE 82 Engineering Mathematics
Class otes : First Order Differential Equation on Linear AE 8 Engineering athematics Universe on Linear umerical Linearization on Linear Special Cases Analtical o General Solution Linear Analtical General
More informationLimits and Continuous Functions. 2.2 Introduction to Limits. We first interpret limits loosely. We write. lim f(x) = L
2 Limits and Continuous Functions 2.2 Introduction to Limits We first interpret limits loosel. We write lim f() = L and sa the limit of f() as approaches c, equals L if we can make the values of f() arbitraril
More informationExercises involving elementary functions
017:11:0:16:4:09 c M K Warby MA3614 Complex variable methods and applications 1 Exercises involving elementary functions 1 This question was in the class test in 016/7 and was worth 5 marks a) Let z +
More informationMath 240 Calculus III
Line Math 114 Calculus III Summer 2013, Session II Monday, July 1, 2013 Agenda Line 1.,, and 2. Line vector Line Definition Let f : X R 3 R be a differentiable scalar function on a region of 3-dimensional
More informationLet F be a field defined on an open region D in space, and suppose that the (work) integral A
16.3 1 16.3 Path Independence and Conservative Fields Definition. Path Independence Let F be a field defined on an open region D in space, and suppose B that the work) integral A F dr is the same for all
More information(2.5) 1. Solve the following compound inequality and graph the solution set.
Intermediate Algebra Practice Final Math 0 (7 th ed.) (Ch. -) (.5). Solve the following compound inequalit and graph the solution set. 0 and and > or or (.7). Solve the following absolute value inequalities.
More informationExact Differential Equations. The general solution of the equation is f x, y C. If f has continuous second partials, then M y 2 f
APPENDIX C Additional Topics in Differential Equations APPENDIX C. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Chapter 6, ou studied applications
More information14.5 The Chain Rule. dx dt
SECTION 14.5 THE CHAIN RULE 931 27. w lns 2 2 z 2 28. w e z 37. If R is the total resistance of three resistors, connected in parallel, with resistances,,, then R 1 R 2 R 3 29. If z 5 2 2 and, changes
More information1. Partial Fraction Expansion All the polynomials in this note are assumed to be complex polynomials.
Partial Fraction Expansion All the polynomials in this note are assumed to be complex polynomials A rational function / is a quotient of two polynomials P, Q with 0 By Fundamental Theorem of algebra, =
More informationVector calculus MA2VC MA3VC Solutions of the review exercises of Section 1.7
Vector calculus MAVC MA3VC 4 5 Solutions of the review eercises of Section.7 Consider the following scalar and vector fields: f = z 3, g = cos+sin +, G = e z î+ ĵ, h = e cos, l = r 3, F = +zî++zĵ++ˆk,
More informationMATH 417 Homework 4 Instructor: D. Cabrera Due July 7. z c = e c log z (1 i) i = e i log(1 i) i log(1 i) = 4 + 2kπ + i ln ) cosz = eiz + e iz
MATH 47 Homework 4 Instructor: D. abrera Due July 7. Find all values of each expression below. a) i) i b) cos i) c) sin ) Solution: a) Here we use the formula z c = e c log z i) i = e i log i) The modulus
More informationNATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part I
NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Comple Analsis II Lecture Notes Part I Chapter 1 Preliminar results/review of Comple Analsis I These are more detailed notes for the results
More informationExact Equations. M(x,y) + N(x,y) y = 0, M(x,y) dx + N(x,y) dy = 0. M(x,y) + N(x,y) y = 0
Eact Equations An eact equation is a first order differential equation that can be written in the form M(, + N(,, provided that there eists a function ψ(, such that = M (, and N(, = Note : Often the equation
More information4.3 Division of Polynomials
4.3 Division of Polynomials Learning Objectives Divide a polynomials by a monomial. Divide a polynomial by a binomial. Rewrite and graph rational functions. Introduction A rational epression is formed
More informationGreen s Theorem Jeremy Orloff
Green s Theorem Jerem Orloff Line integrals and Green s theorem. Vector Fields Vector notation. In 8.4 we will mostl use the notation (v) = (a, b) for vectors. The other common notation (v) = ai + bj runs
More informationW i n t e r r e m e m b e r t h e W O O L L E N S. W rite to the M anageress RIDGE LAUNDRY, ST. H E LE N S. A uction Sale.
> 7? 8 «> ««0? [ -! ««! > - ««>« ------------ - 7 7 7 = - Q9 8 7 ) [ } Q ««
More informationr/lt.i Ml s." ifcr ' W ATI II. The fnncrnl.icniccs of Mr*. John We mil uppn our tcpiiblicnn rcprc Died.
$ / / - (\ \ - ) # -/ ( - ( [ & - - - - \ - - ( - - - - & - ( ( / - ( \) Q & - - { Q ( - & - ( & q \ ( - ) Q - - # & - - - & - - - $ - 6 - & # - - - & -- - - - & 9 & q - / \ / - - - -)- - ( - - 9 - - -
More informationLecture 04. Curl and Divergence
Lecture 04 Curl and Divergence UCF Curl of Vector Field (1) F c d l F C Curl (or rotor) of a vector field a n curlf F d l lim c s s 0 F s a n C a n : normal direction of s follow right-hand rule UCF Curl
More informationSystems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
More informationSOLVING QUADRATICS. Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources
SOLVING QUADRATICS Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources SOLVING QUADRATICS General Form: y a b c Where a, b and c are constants To solve a quadratic equation, the equation
More informationLimits and Their Properties
Chapter 1 Limits and Their Properties Course Number Section 1.1 A Preview of Calculus Objective: In this lesson you learned how calculus compares with precalculus. I. What is Calculus? (Pages 42 44) Calculus
More informationA L T O SOLO LOWCLL. MICHIGAN, THURSDAY. DECEMBER 10,1931. ritt. Mich., to T h e Heights. Bos" l u T H I S COMMl'NiTY IN Wilcox
G 093 < 87 G 9 G 4 4 / - G G 3 -!! - # -G G G : 49 q» - 43 8 40 - q - z 4 >» «9 0-9 - - q 00! - - q q!! ) 5 / : \ 0 5 - Z : 9 [ -?! : ) 5 - - > - 8 70 / q - - - X!! - [ 48 - -!
More informationPoles, Residues, and All That
hapter Ten Poles, Residues, and All That 0.. Residues. A point z 0 is a singular point of a function f if f not analytic at z 0, but is analytic at some point of each neighborhood of z 0. A singular point
More information11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR
SECTION 11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 633 wit speed v o along te same line from te opposite direction toward te source, ten te frequenc of te sound eard b te observer is were c is
More informationTopic 3 Notes Jeremy Orloff
Topic 3 Notes Jerem Orloff 3 Line integrals and auch s theorem 3.1 Introduction The basic theme here is that comple line integrals will mirror much of what we ve seen for multivariable calculus line integrals.
More informationBasics Concepts and Ideas First Order Differential Equations. Dr. Omar R. Daoud
Basics Concepts and Ideas First Order Differential Equations Dr. Omar R. Daoud Differential Equations Man Phsical laws and relations appear mathematicall in the form of Differentia Equations The are one
More informationVCAA Bulletin. VCE, VCAL and VET. Supplement 2
VCE, VCAL and VET VCAA Bulletin Regulations and information about curriculum and assessment for the VCE, VCAL and VET No. 87 April 20 Victorian Certificate of Education Victorian Certificate of Applied
More informationMath RE - Calculus I Functions Page 1 of 10. Topics of Functions used in Calculus
Math 0-03-RE - Calculus I Functions Page of 0 Definition of a function f() : Topics of Functions used in Calculus A function = f() is a relation between variables and such that for ever value onl one value.
More informationUNIVERSIDAD CARLOS III DE MADRID MATHEMATICS II EXERCISES (SOLUTIONS )
UNIVERSIDAD CARLOS III DE MADRID MATHEMATICS II EXERCISES (SOLUTIONS ) CHAPTER : Limits and continuit of functions in R n. -. Sketch the following subsets of R. Sketch their boundar and the interior. Stud
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 information2 Particle dynamics. The particle is called free body if it is no in interaction with any other body or field.
Mechanics and Thermodynamics GEIT5 Particle dynamics. Basic concepts The aim of dynamics is to find the cause of motion and knowing the cause to give description of motion. The cause is always some interaction.
More informationParticular Solutions
Particular Solutions Our eamples so far in this section have involved some constant of integration, K. We now move on to see particular solutions, where we know some boundar conditions and we substitute
More informationTopic 4 Notes Jeremy Orloff
Topic 4 Notes Jeremy Orloff 4 auchy s integral formula 4. Introduction auchy s theorem is a big theorem which we will use almost daily from here on out. Right away it will reveal a number of interesting
More informationDirectional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables.
Directional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables. Partial derivatives and directional derivatives. Directional derivative of functions of
More informationVector Calculus. Vector Fields. Reading Trim Vector Fields. Assignment web page assignment #9. Chapter 14 will examine a vector field.
Vector alculus Vector Fields Reading Trim 14.1 Vector Fields Assignment web page assignment #9 hapter 14 will eamine a vector field. For eample, if we eamine the temperature conditions in a room, for ever
More informationIntegrals along a curve in space. (Sect. 16.1)
Integrals along a curve in space. (Sect. 6.) Line integrals in space. The addition of line integrals. ass and center of mass of wires. Line integrals in space Definition The line integral of a function
More informationDIFFERENTIATION. 3.1 Approximate Value and Error (page 151)
CHAPTER APPLICATIONS OF DIFFERENTIATION.1 Approimate Value and Error (page 151) f '( lim 0 f ( f ( f ( f ( f '( or f ( f ( f '( f ( f ( f '( (.) f ( f '( (.) where f ( f ( f ( Eample.1 (page 15): Find
More informationv t t t t a t v t d dt t t t t t 23.61
SECTION 4. MAXIMUM AND MINIMUM VALUES 285 The values of f at the endpoints are f 0 0 and f 2 2 6.28 Comparing these four numbers and using the Closed Interval Method, we see that the absolute minimum value
More informationx c x c This suggests the following definition.
110 Chapter 1 / Limits and Continuit 1.5 CONTINUITY A thrown baseball cannot vanish at some point and reappear someplace else to continue its motion. Thus, we perceive the path of the ball as an unbroken
More informationMath 421 Midterm 2 review questions
Math 42 Midterm 2 review questions Paul Hacking November 7, 205 () Let U be an open set and f : U a continuous function. Let be a smooth curve contained in U, with endpoints α and β, oriented from α to
More informationOne Solution Two Solutions Three Solutions Four Solutions. Since both equations equal y we can set them equal Combine like terms Factor Solve for x
Algebra Notes Quadratic Systems Name: Block: Date: Last class we discussed linear systems. The only possibilities we had we 1 solution, no solution or infinite solutions. With quadratic systems we have
More informationUS01CMTH02 UNIT-3. exists, then it is called the partial derivative of f with respect to y at (a, b) and is denoted by f. f(a, b + b) f(a, b) lim
Study material of BSc(Semester - I US01CMTH02 (Partial Derivatives Prepared by Nilesh Y Patel Head,Mathematics Department,VPand RPTPScience College 1 Partial derivatives US01CMTH02 UNIT-3 The concept of
More informationu = 0; thus v = 0 (and v = 0). Consequently,
MAT40 - MANDATORY ASSIGNMENT #, FALL 00; FASIT REMINDER: The assignment must be handed in before 4:30 on Thursday October 8 at the Department of Mathematics, in the 7th floor of Niels Henrik Abels hus,
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 informationMath Theory of Number Homework 1
Math 4050 Theory of Number Homework 1 Due Wednesday, 015-09-09, in class Do 5 of the following 7 problems. Please only attempt 5 because I will only grade 5. 1. Find all rational numbers and y satisfying
More informationis the two-dimensional curl of the vector field F = P, Q. Suppose D is described by a x b and f(x) y g(x) (aka a type I region).
Math 55 - Vector alculus II Notes 4.4 Green s Theorem We begin with Green s Theorem: Let be a positivel oriented (parameterized counterclockwise) piecewise smooth closed simple curve in R and be the region
More informationMathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition
Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities
More information2.1 Rates of Change and Limits AP Calculus
. Rates of Change and Limits AP Calculus. RATES OF CHANGE AND LIMITS Limits Limits are what separate Calculus from pre calculus. Using a it is also the foundational principle behind the two most important
More informationTwo Posts to Fill On School Board
Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83
More informationConstant no variables, just a number. Linear Note: Same form as f () x mx b. Quadratic Note: Same form as. Cubic x to the third power
Precalculus Notes: Section. Modeling High Degree Polnomial Functions Graphs of Polnomials Polnomial Notation f ( ) a a a... a a a is a polnomial function of degree n. n n 1 n n n1 n 1 0 n is the degree
More informationPolynomial and Rational Functions
Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define
More informationHigher. Polynomials and Quadratics. Polynomials and Quadratics 1
Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities
More informationWritten Assignment #1 - SOLUTIONS
Written Assignment #1 - SOLUTIONS Question 1. Use the definitions of continuity and differentiability to prove that the function { cos(1/) if 0 f() = 0 if = 0 is continuous at 0 but not differentiable
More informationy=1/4 x x=4y y=x 3 x=y 1/3 Example: 3.1 (1/2, 1/8) (1/2, 1/8) Find the area in the positive quadrant bounded by y = 1 x and y = x3
Eample: 3.1 Find the area in the positive quadrant bounded b 1 and 3 4 First find the points of intersection of the two curves: clearl the curves intersect at (, ) and at 1 4 3 1, 1 8 Select a strip at
More informationUniversity of Regina Department of Mathematics and Statistics
u z v 1. Consider the map Universit of Regina Department of Mathematics and Statistics MATH431/831 Differential Geometr Winter 2014 Homework Assignment No. 3 - Solutions ϕ(u, v) = (cosusinv, sin u sin
More informationName: Student ID: Math 314 (Calculus of Several Variables) Exam 3 May 24 28, Instructions:
Name: Student ID: Section: Instructor: _00 Scott Glasgow Math 314 (Calculus of Several Variables) RED Eam 3 Ma 4 8, 013 Instructions: For questions which require a written answer, show all our work. Full
More informationMath 233. Practice Problems Chapter 15. i j k
Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed
More 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 informationWhat is the limit of a function? Intuitively, we want the limit to say that as x gets closer to some value a,
Limits The notion of a limit is fundamental to the stud of calculus. It is one of the primar concepts that distinguishes calculus from mathematical subjects that ou saw prior to calculus, such as algebra
More informationM 340L CS Homework Set 1
M 340L CS Homework Set 1 Solve each system in Problems 1 6 by using elementary row operations on the equations or on the augmented matri. Follow the systematic elimination procedure described in Lay, Section
More informationSOME PROBLEMS YOU SHOULD BE ABLE TO DO
OME PROBLEM YOU HOULD BE ABLE TO DO I ve attempted to make a list of the main calculations you should be ready for on the exam, and included a handful of the more important formulas. There are no examples
More information3, 1, 3 3, 1, 1 3, 1, 1. x(t) = t cos(πt) y(t) = t sin(πt), z(t) = t 2 t 0
Math 5 Final Eam olutions ecember 5, Problem. ( pts.) (a 5 pts.) Find the distance from the point P (,, 7) to the plane z +. olution. We can easil find a point P on the plane b choosing some values for
More information11.6 Directional Derivative & The Gradient Vector. Working Definitions
1 Ma 016 1 Kidogchi Kenneth The directional derivative o at 0 0 ) in the direction o the nit vector is the scalar nction deined b: where q is angle between the two vectors placed tail-to-tail and 0 < q
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 informationQUADRATIC EQUATIONS. + 6 = 0 This is a quadratic equation written in standard form. x x = 0 (standard form with c=0). 2 = 9
QUADRATIC EQUATIONS A quadratic equation is always written in the form of: a + b + c = where a The form a + b + c = is called the standard form of a quadratic equation. Eamples: 5 + 6 = This is a quadratic
More informationChapter Six. More Integration
hapter Six More Integration 6.. auchy s Integral Formula. Suppose f is analytic in a region containing a simple closed contour with the usual positive orientation, and suppose z is inside. Then it turns
More informationEEE 203 COMPLEX CALCULUS JANUARY 02, α 1. a t b
Comple Analysis Parametric interval Curve 0 t z(t) ) 0 t α 0 t ( t α z α ( ) t a a t a+α 0 t a α z α 0 t a α z = z(t) γ a t b z = z( t) γ b t a = (γ, γ,..., γ n, γ n ) a t b = ( γ n, γ n,..., γ, γ ) b
More informationMA123, Chapter 1: Equations, functions and graphs (pp. 1-15)
MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand
More informationSPS Mathematical Methods
SPS 2281 - Mathematical Methods Assignment No. 2 Deadline: 11th March 2015, before 4:45 p.m. INSTRUCTIONS: Answer the following questions. Check our answer for odd number questions at the back of the tetbook.
More informationLimits and the derivative function. Limits and the derivative function
The Velocity Problem A particle is moving in a straight line. t is the time that has passed from the start of motion (which corresponds to t = 0) s(t) is the distance from the particle to the initial position
More informationMultivariable Calculus Lecture #13 Notes. in each piece. Then the mass mk. 0 σ = σ = σ
Multivariable Calculus Lecture #1 Notes In this lecture, we ll loo at parameterization of surfaces in R and integration on a parameterized surface Applications will include surface area, mass of a surface
More informationLie Symmetry Analysis and Approximate Solutions for Non-linear Radial Oscillations of an Incompressible Mooney Rivlin Cylindrical Tube
Ž Journal of Mathematical Analysis and Applications 45, 34639 doi:116jmaa6748, available online at http:wwwidealibrarycom on Lie Symmetry Analysis and Approimate Solutions for Non-linear Radial Oscillations
More informationChapter XX: 1: Functions. XXXXXXXXXXXXXXX <CT>Chapter 1: Data representation</ct> 1.1 Mappings
978--08-8-8 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Chapter XX: : Functions XXXXXXXXXXXXXXX Chapter : Data representation This section will show you how to: understand
More informationSection 3.3 Limits Involving Infinity - Asymptotes
76 Section. Limits Involving Infinity - Asymptotes We begin our discussion with analyzing its as increases or decreases without bound. We will then eplore functions that have its at infinity. Let s consider
More informationVocabulary. Term Page Definition Clarifying Example degree of a monomial. degree of a polynomial. end behavior. leading coefficient.
CHAPTER 6 Vocabular The table contains important vocabular terms from Chapter 6. As ou work through the chapter, fill in the page number, definition, and a clarifing eample. Term Page Definition Clarifing
More informationAnnouncements. Topics: Homework: - section 12.4 (cross/vector product) - section 12.5 (equations of lines and planes)
Topics: Announcements - section 12.4 (cross/vector product) - section 12.5 (equations of lines and planes) Homework: ü review lecture notes thoroughl ü work on eercises from the tetbook in sections 12.4
More informationACCUPLACER MATH 0311 OR MATH 0120
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises
More informationUNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x
5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able
More informationCoordinates for Projective Planes
Chapter 8 Coordinates for Projective Planes Math 4520, Fall 2017 8.1 The Affine plane We now have several eamples of fields, the reals, the comple numbers, the quaternions, and the finite fields. Given
More informationExample 25: Determine the moment M AB produced by force F in Figure which tends to rotate the rod about the AB axis.
Eample 25: Determine the moment M AB produced by force F in Figure which tends to rotate the rod about the AB ais. Solution: Because that F is parallel to the z-ais so it has no moment about z-ais. Its
More information12/19/2009, FINAL PRACTICE VII Math 21a, Fall Name:
12/19/29, FINAL PRATIE VII Math 21a, Fall 29 Name: MWF 9 Jameel Al-Aidroos MWF 1 Andrew otton-lay MWF 1 Oliver Knill MWF 1 HT Yau MWF 11 Ana araiani MWF 11 hris Phillips MWF 11 Ethan Street MWF 12 Toby
More informationHigher. Differentiation 28
Higher Mathematics UNIT OUTCOME Differentiation Contents Differentiation 8 Introduction to Differentiation 8 Finding the Derivative 9 Differentiating with Respect to Other Variables 4 Rates of Change 4
More informationAPPM 2350 Section Exam points Wednesday October 24, 6:00pm 7:30pm, 2018
APPM 250 Section Eam 2 40 points Wednesda October 24, 6:00pm 7:0pm, 208 ON THE FRONT OF YOUR BLUEBOOK write: () our name, (2) our student ID number, () lecture section/time (4) our instructor s name, (5)
More informationPhysics 217, Fall September 2002
Toda in Phsics 217: vector derivatives First derivatives: Gradient ( ) Divergence ( ) Curl ( ) Second derivatives: the Laplacian ( 2 ) and its relatives Vector-derivative identities: relatives of the chain
More informationExercises involving elementary functions
017:11:0:16:4:09 c M. K. Warby MA3614 Complex variable methods and applications 1 Exercises involving elementary functions 1. This question was in the class test in 016/7 and was worth 5 marks. a) Let
More informationTopic 5.1: Line Element and Scalar Line Integrals
Math 275 Notes Topic 5.1: Line Element and Scalar Line Integrals Textbook Section: 16.2 More Details on Line Elements (vector dr, and scalar ds): http://www.math.oregonstate.edu/bridgebook/book/math/drvec
More informationFixed-Point Iterations (2.2)
Fied-Point Iterations (.). Mean Value Theorem: a. Rolle s Theorem : Suppose that f is continuous on a, b and is differentiable on a, b.iff a f b, then there eists a number c in a, b such that f c. b. Mean
More informationLimits: How to approach them?
Limits: How to approach them? The purpose of this guide is to show you the many ways to solve it problems. These depend on many factors. The best way to do this is by working out a few eamples. In particular,
More informationsummation of series: sommerfeld-watson transformation
Phsics 2400 Spring 207 summation of series: sommerfeld-watson transformation spring semester 207 http://www.phs.uconn.edu/ rozman/courses/p2400_7s/ Last modified: Ma 3, 207 Sommerfeld-Watson transformation,
More informationMath 120 A Midterm 2 Solutions
Math 2 A Midterm 2 Solutions Jim Agler. Find all solutions to the equations tan z = and tan z = i. Solution. Let α be a complex number. Since the equation tan z = α becomes tan z = sin z eiz e iz cos z
More informationSOLVING EQUATIONS OF ONE VARIABLE
1 SOLVING EQUATIONS OF ONE VARIABLE ELM1222 Numerical Analysis Some of the contents are adopted from Laurene V. Fausett, Applied Numerical Analysis using MATLAB. Prentice Hall Inc., 1999 2 Today s lecture
More information18 Equilibrium equation of a laminated plate (a laminate)
z ν = 1 ν = ν = 3 ν = N 1 ν = N z 0 z 1 z N 1 z N z α 1 < 0 α > 0 ν = 1 ν = ν = 3 Such laminates are often described b an orientation code [α 1 /α /α 3 /α 4 ] For eample [0/-45/90/45/0/0/45/90/-45/0] Short
More informationMATH 1010E University Mathematics Lecture Notes (week 8) Martin Li
MATH 1010E University Mathematics Lecture Notes (week 8) Martin Li 1 L Hospital s Rule Another useful application of mean value theorems is L Hospital s Rule. It helps us to evaluate its of indeterminate
More informationLecture 5: Finding limits analytically Simple indeterminate forms
Lecture 5: Finding its analytically Simple indeterminate forms Objectives: (5.) Use algebraic techniques to resolve 0/0 indeterminate forms. (5.) Use the squeeze theorem to evaluate its. (5.3) Use trigonometric
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More information