Section 5: The Jacobian matrix and applications. S1: Motivation S2: Jacobian matrix + differentiability S3: The chain rule S4: Inverse functions


 Caroline Hardy
 1 years ago
 Views:
Transcription
1 Section 5: The Jacobian matri and applications. S1: Motivation S2: Jacobian matri + differentiabilit S3: The chain rule S4: Inverse functions Images from Thomas calculus b Thomas, Wier, Hass & Giordano, 2008, Pearson Education, Inc. 1
2 S1: Motivation. Our main aim of this section is to consider general functions and to define a general derivative and to look at its properties. In fact, we have slowl been doing this. We first considered vector valued functions of one variable f : R R n and defined the derivative as f(t) (f 1 (t),..., f n (t)) f (t) (f 1 (t),..., f n(t)). We then considered real valued functions of two and three variables f : R 2 R, f : R 3 R and (as we will see later) we ma think of the derivatives of these functions, respectivel, as f ( f/, f/ ) f ( f/, f/, f/ z). 2
3 There are still more general functions than those two or three tpes above. If we combine the elements of each, then we can form vector valued functions of man variables. A function f : R m R n (n > 1) is a vector valued function of m variables. Eample 1 f z ( + + z z ) defines a function from R 3 to R 2. 3
4 When it comes to these vector valued functions, we should write vectors as column vectors (essentiall because matrices act on column vectors), however, we will use both vertical columns and horizontal m tuple notation. Thus, for eample, for the vector R 3 we will write both or (,, z) (and i + j + zk) z and so we could write f : R 3 R 2 as f1 (,, z) f and f z 2 (,, z) f(,, z) (f 1 (,, z), f 2 (,, z)) f 1 (,, z)i + f 2 (,, z)j or combinations of columns and mtuples. 4
5 In Eample 1, the real valued functions f 1 f z and z z z are called the co ordinate or component functions of f, and we ma write f1 f. f 2 Generall, an f : R m R n is determined b n co ordinate functions f 1,..., f n and we write f f 1 ( 1,..., m ) f 2 ( 1,..., m ).. (1) f n ( 1,..., m ) 5
6 We shall be most interested in the cases where f : R 2 R 2 or f : R 3 R 3 because this is where the most applications occur and because it will prove to be etremel useful in our topic on multiple integration. For these special cases we can use the following notation f() f(, ) (f 1 (, ), f 2 (, )) f 1 (, )i + f 2 (, )j. f() f(,, z) (f 1 (,, z), f 2 (,, z), f 3 (,, z)) f 1 (,, z)i + f 2 (,, z)j + f 3 (,, z)k. 6
7 One wa of visualizing f, sa, f : R 2 R 2 is to think of f as a transformation between co ordinate planes. So that f ma stretch, compress, rotate etc sets in its domain. The above be particularl useful when dealing with multiple integration and change of variables. 7
8 S2: Jacobian matri + differentiabilit. Our first problem is how we define the derivative of a vector valued function of man variables. Recall that if f : R 2 R then we can form the directional derivative, i.e., D u f u 1 f + u 2 f f u where u (u 1, u 2 ) is a unit vector. Thus, knowledge of the gradient of f gives information about all directional derivatives. Therefore it is reasonable to assume p f ( f f (p), (p) is the derivative of f at p. (The stor is more complicated than this but when we sa f is differentiable we mean f represents the derivative, to be discussed a little later.) ) 8
9 More generall if f : R m R then we take the derivative at p to be the row vector ( f (p), f ) f (p),..., (p) p f 1 2 m Now take f : R m R n where f is as in equation (1), then the natural candidate for the derivative of f at p is f 1 f f 1 m f 2 f 2 f J p f m f n f n f n m where the partial derivatives are evaluated at p. This n m matri is called the Jacobian matri of f. Writing the function f as a column helps us to get the rows and columns of the Jacobian matri the right wa round. Note the Jacobian is usuall the determinant of this matri when the matri is square, i.e., when m n. 9
10 Eample 2 Find the Jacobian matri of f from Eample 1 and evaluate it at (1, 2, 3). 10
11 Most of the cases we will be looking at have m n either 2 or 3. Suppose u u(, ) and v v(, ). If we define f : R 2 R 2 b f then the Jacobian matri is Jf and the Jacobian (determinant) det(jf) u v We often denote det(jf) b u(, ) v(, ) u v u v (u, v) u v u f1 f 2 v v u. (, ). 11
12 Eample 3 Consider the transformation from polar to Cartesian co ordinates, where r cos θ and r sin θ. We have (, ) (r, θ) r r θ θ cos θ sin θ r sin θ r cos θ r. 12
13 We have alread noted that if f : R m R n then the Jacobian matri at each point a R m is an m n matri. Such a matri J a f gives us a linear map D a f : R m R n defined b (D a f) () : J a f for all R n. Note that is a column vector. When we sa f : R m R n is differentiable at q we mean that, the affine function A() : f(q) + ( J q f ) ( q), is a good approimation to f() near q in the sense that where lim q q f() f(q) (J q f) ( q) q 0 ( 1 q 1 ) ( m q m ) 2. 13
14 You should compare this to the one variable case: a function f : R R f(a + h) f(a) is differentiable at a if lim eists, and we call this h 0 h limit f (a). But we could equall well sa this as f : R R is differentiable at a if there is a number, written f (a), for which lim h 0 f(a + h) f(a) f (a) h h 0, because a linear map L : R R can onl operate b multiplication with a number. How do we easil recognize a differentiable function? If all of the component functions of the Jacobian matri of f are continuous, then f is differentiable. 14
15 Eample 4 Write the derivative of the function in Eample 1 at (1, 2, 3) as a linear map. Suppose f and g are two differentiable functions from R m to R n. It is eas to see that the derivative of f +g is the sum of the derivatives of f and g. We can take the dot product of f and g and get a function from R m to R, and then differentiate that. The result is a sort of product rule, but I ll leave ou to work out what happens. Since we cannot divide vectors, there cannot be a quotient rule, so of the standard differentiation rules, that leaves the chain rule. 15
16 S3: The chain rule. Now suppose that g : R m R s and f : R s R n. We can now form the composition f g b mapping with g first and then following with f: g() f(g()) (2) (f g) () : f (g()) for all R m. Eample 5 Let g : R 2 R 2 and f : R 2 R 3 be defined, respectivel, b g : + Then f g is defined b (f g) f ( g ) and f f + : sin. sin( + ) + ( + ) (). 16
17 Let b g(p) R s. If f and g in (2) above are differentiable then the maps J p g : R m R s and J b f : R s R n are defined, and we have the following general result. Theorem 1 (The Chain Rule) Suppose that g : R m R s and f : R s R n are differentiable. Then J p (f g) J g(p) f J p g. This is again just like the one variable case, ecept now we are multipling matrices (see below). 17
18 Eample 6 Consider Eample 5: g + and f sin. Find J p (f g) where p ( a1 Also J p g J g(p) f ) a 2 ) ( 1 1. We have p cos ( 1 1 a 2 a 1 ) a 1 +a 2,a 1 a 2 cos(a 1 + a 2 ) a 1 a 2 a 1 + a
19 (E cont.) and We observe that J p (f g) cos( + ) cos( + ) cos(a 1 + a 2 ) cos(a 1 + a 2 ) 1 a 2 2a 1 a 2 + a a 1 a a 1 a 2 cos(a 1 + a 2 ) a 1 a 2 a 1 + a 2 ( 1 1 a 2 a 1 ) p 19
20 The one variable chain rule is a special case of the chain rule that we ve just met the same can be said for the chain rules we saw in earlier sections. Let : R R be a differentiable function of t and and let u : R R a differentiable function of. Then (u ) : R R is given b (u )(t) u((t)). In the notation of this chapter i.e. J t (u ) J (t) u J t ] [ ] [ du d (u ) dt d dt [ d t (t) ] t. We usuall write this as du dt du d d dt keeping in mind that when we write du we are thinking of u as a dt function of t, i.e., u((t)) and when we write du we are thinking of u d as a function of. 20
21 Now suppose we have (t), (t) and z f(, ). Then J t (f ) J (t) f J t Therefore so that d dt (f((t), (t))) df dt f ( f d dt + f which is just what we saw in earlier sections. f d dt, ) d dt d dt 21
22 S4: Inverse functions. In first ear (or earlier) ou will have met the inverse function theorem, which sas essentiall that if f (a) is not zero, then there is a differentiable inverse function f 1 defined near f(a) with [ d dt (f 1 ) ] f(a) 1 f (a). What happens in the multi variable case? 22
23 Let us consider a case where we can write down the inverse. For polar coordinates we have r cos θ, r 2 + 2, Now differentiating we obtain r r cos θ r r sin θ θ arctan cos θ and. r cos θ r i.e., 1. r We see that the one variable inverse function theorem does not appl to partial derivatives. However, there is a simple generalisation if we use the multivariable derivative, that is, the Jacobian matri. 23
24 To continue with the polar coordinate eample, define r (r, θ) r cos θ f θ (r, θ) r sin θ (3) and Consider (f g) g f ( r(, ) θ(, ) g ) f arctan. (4) r θ Id. Therefore f g Id, the identit operator on R 2. Similarl g f Id. Recall ( Id ) so that J(Id) identit matri. 24
25 Thus b the chain rule Jf Jg J(Id) Jg Jf so that (Jf) 1 Jg. Note for simplicit the points of evaluation have been left out. Therefore r θ r θ 1 r r θ We can check this directl b substituting r etc. θ cos θ The same idea works in general: 25
26 Theorem 2 (The Inverse Function Theorem) Let f : R n R n be differentiable at p. If J p f is an invertible matri then there is an inverse function f 1 : R n R n defined in some neighbourhood of b f(p) and (J b f 1 ) (J p f) 1. Note that the inverse function ma onl eist in a small region around b f(p). Eample 7 We earlier saw that for polar coordinates, with the notation of equation (3) cos θ r sin θ Jf, sin θ r cos θ with determinant r. So it follows from the inverse function theorem that the inverse function g is differentiable if r 0. 26
27 Eample 8 The function f : R 2 R 2 is given b ( u f 2 2 ) v Where is f invertible? invertible. ( 2 2 SOLN: Jf 2 2 everwhere ecept the aes. Find the Jacobian matri of f 1 where f is 1 Jf ( ) and det Jf 8, so f is invertible ) 1 4 Translate to (u, v) coordinates and this is ( Jf 1 2 (u + v) 1/2 (u + v) 1/2 4 (v u) 1/2 (v u) 1/2 ( ). 1 ). 27
28 Finall let us appl the inverse function theorem to the Jacobian determinants. We recall that (r, θ) (, ) (, ) (r, θ) det Jg det Jf r θ r r r θ θ. θ and Since Jg and Jf are inverse matrices, their determinants are inverses: (r, θ) (, ) 1 (,) (r,θ) This sort of result is true for an change of variable in an number of dimensions and will prove ver useful in integration.. 28
MAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More informationf x a 0 n 1 a 0 a 1 cos x a 2 cos 2x a 3 cos 3x b 1 sin x b 2 sin 2x b 3 sin 3x a n cos nx b n sin nx n 1 f x dx y
Fourier Series When the French mathematician Joseph Fourier (768 83) was tring to solve a problem in heat conduction, he needed to epress a function f as an infinite series of sine and cosine functions:
More informationDownloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x
Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its
More informationCOMPONENTS OF VECTORS
COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationGLOBAL COORDINATE METHOD FOR DETERMINING SENSITIVITY IN ASSEMBLY TOLERANCE ANALYSIS
GOBA COORDINATE METOD FOR DETERMINING SENSITIVIT IN ASSEMB TOERANCE ANASIS Jinsong Gao ewlettpackard Corp. InkJet Business Unit San Diego, CA Kenneth W. Chase Spencer P. Magleb Mechanical Engineering
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two or threedimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationPROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS
PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationy 1 x dx ln x y a x dx 3. y e x dx e x 15. y sinh x dx cosh x y cos x dx sin x y csc 2 x dx cot x 7. y sec 2 x dx tan x 9. y sec x tan x dx sec x
Strateg for Integration As we have seen, integration is more challenging than differentiation. In finding the derivative of a function it is obvious which differentiation formula we should appl. But it
More informationLecture 1 Introduction 1. 1.1 Rectangular Coordinate Systems... 1. 1.2 Vectors... 3. Lecture 2 Length, Dot Product, Cross Product 5. 2.1 Length...
CONTENTS i Contents Lecture Introduction. Rectangular Coordinate Sstems..................... Vectors.................................. 3 Lecture Length, Dot Product, Cross Product 5. Length...................................
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationVector Fields and Line Integrals
Vector Fields and Line Integrals 1. Match the following vector fields on R 2 with their plots. (a) F (, ), 1. Solution. An vector, 1 points up, and the onl plot that matches this is (III). (b) F (, ) 1,.
More informationPrecalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES
Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 01 Sets There are no statemandated Precalculus 02 Operations
More informationChapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation
Chapter 6 Linear Transformation 6 Intro to Linear Transformation Homework: Textbook, 6 Ex, 5, 9,, 5,, 7, 9,5, 55, 57, 6(a,b), 6; page 7 In this section, we discuss linear transformations 89 9 CHAPTER
More information15.1. Exact Differential Equations. Exact FirstOrder Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact FirstOrder Equations 09 SECTION 5. Eact FirstOrder Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationSECTION 51 Exponential Functions
354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3D We have defined the velocit and acceleration of a particle as the first and second
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the ais and the tangent
More informationVersion 005 Exam Review Practice Problems NOT FOR A GRADE alexander (55715) 1. Hence
Version 005 Eam Review Practice Problems NOT FOR A GRADE aleander 5575 This printout should have 47 questions Multiplechoice questions may continue on the net column or page find all choices before answering
More informationMathematics Placement Packet Colorado College Department of Mathematics and Computer Science
Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking
More informationCHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS
CHAPTER 0 SYSTEMS, MATRICES, AND DETERMINANTS PRECALCULUS: A TEACHING TEXTBOOK Lesson 64 Solving Sstems In this chapter, we re going to focus on sstems of equations. As ou ma remember from algebra, sstems
More informationSAMPLE. Polynomial functions
Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through
More informationWhen I was 3.1 POLYNOMIAL FUNCTIONS
146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationCollege Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More informationCalculus with Parametric Curves
Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationPolynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will
More informationPre Calculus Math 40S: Explained!
Pre Calculus Math 0S: Eplained! www.math0s.com 0 Logarithms Lesson PART I: Eponential Functions Eponential functions: These are functions where the variable is an eponent. The first tpe of eponential graph
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationYou know from calculus that functions play a fundamental role in mathematics.
CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationSimilar matrices and Jordan form
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
More information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
More information13 CALCULUS OF VECTORVALUED FUNCTIONS
CALCULUS OF VECTORVALUED FUNCTIONS. VectorValued Functions LT Section 4.) Preliminar Questions. Which one of the following does not parametrize a line? a) r t) 8 t,t,t b) r t) t i 7t j + t k c) r t)
More informationMATH1231 Algebra, 2015 Chapter 7: Linear maps
MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter
More informationLines and Planes 1. x(t) = at + b y(t) = ct + d
1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b
More informationWHEN DOES A CROSS PRODUCT ON R n EXIST?
WHEN DOES A CROSS PRODUCT ON R n EXIST? PETER F. MCLOUGHLIN It is probably safe to say that just about everyone reading this article is familiar with the cross product and the dot product. However, what
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More informationLINEAR MAPS, THE TOTAL DERIVATIVE AND THE CHAIN RULE. Contents
LINEAR MAPS, THE TOTAL DERIVATIVE AND THE CHAIN RULE ROBERT LIPSHITZ Abstract We will discuss the notion of linear maps and introduce the total derivative of a function f : R n R m as a linear map We will
More informationLecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function
Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between
More informationFind the Relationship: An Exercise in Graphing Analysis
Find the Relationship: An Eercise in Graphing Analsis Computer 5 In several laborator investigations ou do this ear, a primar purpose will be to find the mathematical relationship between two variables.
More informationDear Accelerated PreCalculus Student:
Dear Accelerated PreCalculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, collegepreparatory mathematics course that will also
More informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationcorrectchoice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More informationDomain of a Composition
Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More information3D Stress Components. From equilibrium principles: τ xy = τ yx, τ xz = τ zx, τ zy = τ yz. Normal Stresses. Shear Stresses
3D Stress Components From equilibrium principles:, z z, z z The most general state of stress at a point ma be represented b 6 components Normal Stresses Shear Stresses Normal stress () : the subscript
More informationz 0 and y even had the form
Gaussian Integers The concepts of divisibility, primality and factoring are actually more general than the discussion so far. For the moment, we have been working in the integers, which we denote by Z
More information( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More informationContent. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11
Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter
More informationFunctions and Their Graphs
3 Functions and Their Graphs On a sales rack of clothes at a department store, ou see a shirt ou like. The original price of the shirt was $00, but it has been discounted 30%. As a preferred shopper, ou
More informationModelling musical chords using sine waves
Modelling musical chords using sine waves Introduction From the stimulus word Harmon, I chose to look at the transmission of sound waves in music. As a keen musician mself, I was curious to understand
More informationMATH 132: CALCULUS II SYLLABUS
MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early
More information5.7 Maximum and Minimum Values
5.7 Maximum and Minimum Values Objectives Icandefinecriticalpoints. I know the di erence between local and absolute minimums/maximums. I can find local maximum(s), minimum(s), and saddle points for a given
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationSECTION 22 Straight Lines
 Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above
More informationVector Algebra CHAPTER 13. Ü13.1. Basic Concepts
CHAPTER 13 ector Algebra Ü13.1. Basic Concepts A vector in the plane or in space is an arrow: it is determined by its length, denoted and its direction. Two arrows represent the same vector if they have
More informationis in plane V. However, it may be more convenient to introduce a plane coordinate system in V.
.4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a twodimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5
More informationThis week. CENG 732 Computer Animation. The Display Pipeline. Ray Casting Display Pipeline. Animation. Applying Transformations to Points
This week CENG 732 Computer Animation Spring 20062007 Week 2 Technical Preliminaries and Introduction to Keframing Recap from CEng 477 The Displa Pipeline Basic Transformations / Composite Transformations
More informationName Date. BreakEven Analysis
Name Date BreakEven Analsis In our business planning so far, have ou ever asked the questions: How much do I have to sell to reach m gross profit goal? What price should I charge to cover m costs and
More informationConstrained Optimization: The Method of Lagrange Multipliers:
Constrained Optimization: The Method of Lagrange Multipliers: Suppose the equation p(x,) x 60x 7 00 models profit when x represents the number of handmade chairs and is the number of handmade rockers produced
More informationProduct Operators 6.1 A quick review of quantum mechanics
6 Product Operators The vector model, introduced in Chapter 3, is ver useful for describing basic NMR eperiments but unfortunatel is not applicable to coupled spin sstems. When it comes to twodimensional
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More information1 Local Brouwer degree
1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationAnswer Key for the Review Packet for Exam #3
Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math MaMin Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationPRECALCULUS GRADE 12
PRECALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationChapter 8. Lines and Planes. By the end of this chapter, you will
Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes
More informationCryptography and Network Security. Prof. D. Mukhopadhyay. Department of Computer Science and Engineering. Indian Institute of Technology, Kharagpur
Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 12 Block Cipher Standards
More informationBirmingham City Schools
Activity 1 Classroom Rules & Regulations Policies & Procedures Course Curriculum / Syllabus LTF Activity: Interval Notation (Precal) 2 PreAssessment 3 & 4 1.2 Functions and Their Properties 5 LTF Activity:
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationMath 152, Intermediate Algebra Practice Problems #1
Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationTo define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions
Chapter 7 Functions This unit defines and investigates functions as algebraic objects. First, we define functions and discuss various means of representing them. Then we introduce operations on functions
More informationTransformations and Expectations of random variables
Transformations and Epectations of random variables X F X (): a random variable X distributed with CDF F X. Any function Y = g(x) is also a random variable. If both X, and Y are continuous random variables,
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationLINES AND PLANES CHRIS JOHNSON
LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3space, as well as define the angle between two nonparallel planes, and determine the distance
More informationNAME DATE PERIOD. 11. Is the relation (year, percent of women) a function? Explain. Yes; each year is
 NAME DATE PERID Functions Determine whether each relation is a function. Eplain.. {(, ), (0, 9), (, 0), (7, 0)} Yes; each value is paired with onl one value.. {(, ), (, ), (, ), (, ), (, )}. No; in the
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationChapter 16, Part C Investment Portfolio. Risk is often measured by variance. For the binary gamble L= [, z z;1/2,1/2], recall that expected value is
Chapter 16, Part C Investment Portfolio Risk is often measured b variance. For the binar gamble L= [, z z;1/,1/], recall that epected value is 1 1 Ez = z + ( z ) = 0. For this binar gamble, z represents
More information