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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

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 m-tuples. 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

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 information

Identifying second degree equations

Identifying 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 information

f 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

f 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 information

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Downloaded 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 information

COMPONENTS OF VECTORS

COMPONENTS 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 information

Trigonometry Review Workshop 1

Trigonometry 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 information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS 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 information

GLOBAL COORDINATE METHOD FOR DETERMINING SENSITIVITY IN ASSEMBLY TOLERANCE ANALYSIS

GLOBAL COORDINATE METHOD FOR DETERMINING SENSITIVITY IN ASSEMBLY TOLERANCE ANALYSIS GOBA COORDINATE METOD FOR DETERMINING SENSITIVIT IN ASSEMB TOERANCE ANASIS Jinsong Gao ewlett-packard Corp. InkJet Business Unit San Diego, CA Kenneth W. Chase Spencer P. Magleb Mechanical Engineering

More information

Core Maths C2. Revision Notes

Core 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 information

2.1 Three Dimensional Curves and Surfaces

2.1 Three Dimensional Curves and Surfaces . Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The

More information

PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS

PROPERTIES 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 information

Orthogonal Projections

Orthogonal 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 information

Higher. Polynomials and Quadratics 64

Higher. 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 information

y 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

y 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 information

Lecture 1 Introduction 1. 1.1 Rectangular Coordinate Systems... 1. 1.2 Vectors... 3. Lecture 2 Length, Dot Product, Cross Product 5. 2.1 Length...

Lecture 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 information

A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A 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 information

Vector Fields and Line Integrals

Vector 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 information

Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES

Precalculus 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 0-1 Sets There are no state-mandated Precalculus 0-2 Operations

More information

Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation

Chapter 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 information

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

More information

SECTION 5-1 Exponential Functions

SECTION 5-1 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 information

Physics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal

Physics 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 3-D We have defined the velocit and acceleration of a particle as the first and second

More information

Click here for answers.

Click 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 information

Version 005 Exam Review Practice Problems NOT FOR A GRADE alexander (55715) 1. Hence

Version 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 print-out should have 47 questions Multiple-choice questions may continue on the net column or page find all choices before answering

More information

Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science

Mathematics 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 information

CHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS

CHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS CHAPTER 0 SYSTEMS, MATRICES, AND DETERMINANTS PRE-CALCULUS: 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 information

SAMPLE. Polynomial functions

SAMPLE. 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 information

When I was 3.1 POLYNOMIAL FUNCTIONS

When 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 information

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

CITY 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 information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell 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 information

1 VECTOR SPACES AND SUBSPACES

1 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 information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

More information

Calculus with Parametric Curves

Calculus 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 information

α = u v. In other words, Orthogonal Projection

α = 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 information

Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Polynomials. 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 information

Pre Calculus Math 40S: Explained!

Pre 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 information

Polynomial Degree and Finite Differences

Polynomial 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 information

LINEAR ALGEBRA W W L CHEN

LINEAR 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 information

You know from calculus that functions play a fundamental role in mathematics.

You 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 information

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Recall 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 information

Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors

Adding 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 information

Similar matrices and Jordan form

Similar 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 information

9 MATRICES AND TRANSFORMATIONS

9 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 information

13 CALCULUS OF VECTOR-VALUED FUNCTIONS

13 CALCULUS OF VECTOR-VALUED FUNCTIONS CALCULUS OF VECTOR-VALUED FUNCTIONS. Vector-Valued 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 information

MATH1231 Algebra, 2015 Chapter 7: Linear maps

MATH1231 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 information

Lines and Planes 1. x(t) = at + b y(t) = ct + d

Lines 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 information

WHEN DOES A CROSS PRODUCT ON R n EXIST?

WHEN 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 information

Math 215 HW #6 Solutions

Math 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 information

LINEAR MAPS, THE TOTAL DERIVATIVE AND THE CHAIN RULE. Contents

LINEAR 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 information

Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function

Lecture 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 information

Find the Relationship: An Exercise in Graphing Analysis

Find 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 information

Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also

More information

Chapter 6 Quadratic Functions

Chapter 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 information

Math 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 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 information

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

correct-choice 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 information

Domain of a Composition

Domain 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 information

15.062 Data Mining: Algorithms and Applications Matrix Math Review

15.062 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 information

3D 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: 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 information

z 0 and y even had the form

z 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

( ) 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 information

Content. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11

Content. 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 information

Functions and Their Graphs

Functions 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 information

Modelling musical chords using sine waves

Modelling 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 information

MATH 132: CALCULUS II SYLLABUS

MATH 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 information

5.7 Maximum and Minimum Values

5.7 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 information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

SECTION 2-2 Straight Lines

SECTION 2-2 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 information

Vector Algebra CHAPTER 13. Ü13.1. Basic Concepts

Vector 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 information

is in plane V. However, it may be more convenient to introduce a plane coordinate system in V.

is 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 two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5

More information

This week. CENG 732 Computer Animation. The Display Pipeline. Ray Casting Display Pipeline. Animation. Applying Transformations to Points

This week. CENG 732 Computer Animation. The Display Pipeline. Ray Casting Display Pipeline. Animation. Applying Transformations to Points This week CENG 732 Computer Animation Spring 2006-2007 Week 2 Technical Preliminaries and Introduction to Keframing Recap from CEng 477 The Displa Pipeline Basic Transformations / Composite Transformations

More information

Name Date. Break-Even Analysis

Name Date. Break-Even Analysis Name Date Break-Even 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 information

Constrained Optimization: The Method of Lagrange Multipliers:

Constrained 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 information

Product Operators 6.1 A quick review of quantum mechanics

Product 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 two-dimensional

More information

Section 1.1. Introduction to R n

Section 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 information

Solutions to old Exam 1 problems

Solutions 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 information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear 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 information

ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES 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 information

1 Local Brouwer degree

1 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 information

x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

x1 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 information

Answer Key for the Review Packet for Exam #3

Answer Key for the Review Packet for Exam #3 Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y

More information

A vector is a directed line segment used to represent a vector quantity.

A 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 information

PRE-CALCULUS GRADE 12

PRE-CALCULUS GRADE 12 PRE-CALCULUS 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 information

Chapter 8. Lines and Planes. By the end of this chapter, you will

Chapter 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 information

Cryptography 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 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 information

Birmingham City Schools

Birmingham City Schools Activity 1 Classroom Rules & Regulations Policies & Procedures Course Curriculum / Syllabus LTF Activity: Interval Notation (Precal) 2 Pre-Assessment 3 & 4 1.2 Functions and Their Properties 5 LTF Activity:

More information

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY 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 information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

Math 152, Intermediate Algebra Practice Problems #1

Math 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 information

A Quick Algebra Review

A 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 information

To define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions

To 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 information

Transformations and Expectations of random variables

Transformations 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 information

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero 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 information

LINES AND PLANES CHRIS JOHNSON

LINES AND PLANES CHRIS JOHNSON LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3-space, as well as define the angle between two non-parallel planes, and determine the distance

More information

NAME DATE PERIOD. 11. Is the relation (year, percent of women) a function? Explain. Yes; each year is

NAME 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 information

Lecture 14: Section 3.3

Lecture 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 information

Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.

Recall 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 n-dimensional 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 information

1 Sets and Set Notation.

1 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 information

Review of Intermediate Algebra Content

Review 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 information

Chapter 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 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