# Høgskolen i Narvik Sivilingeniørutdanningen

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Høgskolen i Narvik Sivilingeniørutdanningen Eksamen i Faget STE67 ELEMENTMETODEN Klasse: 4.ID Dato: Tid: Kl Tillatte hjelpemidler under eksamen: Kalkulator. Bok Numerical solution of partial differential equations by the finite element method Bok Eleementmetoder. Forelesningsnotater I, II Forelesningsnotater Engelsk/Norsk, Norsk/Engelsk ordbok Faglig kontaktperson under eksamen: Ekstern Professor Gregory A. Chechkin tel Narvik 23

2 . Consider two-point boundary value problem (D): u (x) = x 3 + x 2 for < x < 2 u () = ; u(2) =. a) Derive the integral identity and explain why the solution u is also the solution of a variational problem (V). Solution. Let us introduce the set of admissible functions in the following way: V = {v C [, 2]; v(2) = }. Multiplying the equation by the test-function v V, integrating over [, 2], we obtain u (x)v(x) dx = (x 3 + x 2 )v(x) dx, and finally integrating by parts and using the fact that v(2) = and u () =, we deduce u (x)v (x) dx u (2)v(2)+u ()v() = which is valid for any function v V. The formulation is: Find u V such that u (x)v (x) dx = (u, v ) = ((x 3 + x 2 ), v) v V, (x 3 +x 2 )v(x) dx, where (f, g) = f(x)g(x) dx. b) Find the functional, formulate the minimization problem and explain why the solution u is also the solution of a minimization problem (M). 2

3 Solution. The formulation is: Find u V such that F(u) F(v) for any v V, where F(v) = 2 (v, v ) ((x 3 + x 2 ), v). c) Prove the equivalence of these formulations, i.e. (M) (V ) (D). Solution. By the tasks a) and b) we proved that (D) = (V ), (D) = (M) Now let us check that (V ) = (M). Assume that u is a solution of (V), let v V and set w = v u so that v = u + w and w V. We have F(v) = F(u + w) = 2 (u + w, u + w ) ((x 3 + x 2 ), u + w) = = 2 (u, u ) ((x 3 + x 2 ), u) + (u, w ) ((x 3 + x 2 ), w) + 2 (w, w ) = = F(u) (w, w ) F(u), since (u, w ) ((x 3 + x 2 ), w) = and (w, w ). Let us show that (M) = (V ). Assume that u is a solution of (M), let v V and denote by g(t) the function g(t) = F(u+tv) = 2 (u, u )+t(u, v )+ t2 2 (v, v ) ((x 3 +x 2 ), u) t((x 3 +x 2 ), v). The differentiable function g(t) has a minimum at t = and hence g () =. 3

4 It is easy to see that and hence u is a solution of (V). Summing up, we have shown that g () = (u, v ) ((x 3 + x 2 ), v) (D) = (V ) (M). Finally, if u is a smooth weak solution, then from the integral identity of (V) integrating by parts in the back direction we can obtain the equation of (D). Everything is proved. d) Define the space V h of piecewise linear functions and show that (u u h ) 2 + ((u u h ) ) 2 dx K (u v) 2 + ((u v) ) 2 dx for any v V h. Here u h is the approximate solution of the respective variational problem (V h ). Solution. Recall that u is a solution of (D) ( respectively (V) ) and u h is a solution of (V h ) that is (u h, v ) = ((x 3 + x 2 ), v) v V h. Subtracting the integral identities for (V) and (V h ) we obtain ((u u h ), v ) = v V h. () Let v V h be an arbitrary function and set w = u h v. Then w V h and using () with v replaced by w, we get, using Cauchy-Schwarz-Bunyakovski s inequality ((u u h ) ) 2 dx = ((u u h ), (u u h ) ) + ((u u h ), w ) = 4

5 Dividing by we obtain = ((u u h ), (u u h + w) ) = ((u u h ), (u v) ) 2 ((u u h ) ) 2 dx ((u v) ) 2 dx ((u u h ) ) 2 dx ((u u h ) ) 2 dx Using the Friedrichs inequality, we get (u u h ) 2 + ((u u h ) ) 2 dx K and, hence, by (2) we deduce (u u h ) 2 + ((u u h ) ) 2 dx K K ((u v) ) 2 dx K 2, 2. ((u v) ) 2 dx. (2) ((u u h ) ) 2 dx ((u u h ) ) 2 dx (u v) 2 + ((u v) ) 2 dx. 2. Explain why v P 3 (, 2) is uniquely determined by the values v(), v (), v(2), v (2). Find the corresponding basis functions. Solution. To determine a cubic polynom, which has the form a 3 x 3 + a 2 x 2 + a x + a, 5

6 in the unique way it is necessary to have four different conditions. Let us write the given conditions and check that they are different. We have a a a + a = v() 3a a 2 + a = v () a a a 2 + a = v(2) 3a a a = v (2) They are different if det A, where A is a matrix of the system. The determinant of the matrix A is equal the the following: = Then there exists only one solution of the system and consequently the cubic polynom is determined uniquely. Let us remind that there exists four basis functions which correspond to the following conditions: and v() =, v () =, v(2) =, v (2) = ; v() =, v () =, v(2) =, v (2) = ; v() =, v () =, v(2) =, v (2) = v() =, v () =, v(2) =, v (2) = ; It is possible to calculate them. They are cubic then they have the form ψ i = a i x 3 + b i x 2 + c i x + d i, i =, 2, 3, 4. Let us find the first function. Consider the system a b c d =.

7 The solution is a = 4, b = 3 4, c =, d = or ψ = 4 x3 3 4 x2 +. Let us find the second function. Consider the system The solution is a 2 b 2 c 2 d 2 =. a 2 = 4, b 2 =, c 2 =, d 2 = or ψ 2 = 4 x3 x 2 + x. Let us find the third function. Consider the system The solution is a 3 b 3 c 3 d 3 =. a 3 = 4, b 3 = 3 4, c 3 =, d 3 = or ψ 3 = 4 x x2. Let us find the fourth function. Consider the system a 4 b 4 c 4 d 4 =.

8 The solution is a 4 = 4, b 4 = 2, c 4 =, d 4 = or ψ 4 = 4 x3 2 x2. 3. Consider piecewise linear finite element space V h with basis elements {ϕ j (x)}. Find the element stiffness matrix for the triangle K with vertices at (, ), (2, ), (, ). Solution. Without loss of generality let us denote by ϕ the finction which is equal to in the point (, ), by ϕ 2 the function which is equal to in the point (2, ) and by ϕ 3 the function which is equal to in the point (, ). The element stiffness matrix has the form where a K (ϕ, ϕ ) a K (ϕ, ϕ 2 ) a K (ϕ, ϕ 3 ) a K (ϕ 2, ϕ ) a K (ϕ 2, ϕ 2 ) a K (ϕ 2, ϕ 3 ) a K (ϕ 3, ϕ ) a K (ϕ 3, ϕ 2 ) a K (ϕ 3, ϕ 3 ) a K (ϕ i, ϕ j ) = ϕ i ϕ j dx. Let us calculate the gradient of each basic functions. In fact they are ( ) ( ) ( ) ϕ = 2, ϕ 2 = 2, ϕ 3 =. Finally the element stiffness matrix is equal to Consider the problem (D) K u (x) = sin x for < x < u () = ; u() =. 8,

9 a) Formulate the problem (V h ), which corresponds to problem (D) in terms of stiffness matrix, load vector and coefficients of unknown function. Solution. The variational formulation in V h is (V h ) Find u h V h : (u h, v ) = (sin x, v) v V h. It is easy to prove that instead of considering this identity for any v V h one can consider only M equations with basic functions (u h, ϕ ) = (sin x, ϕ ) (u h, ϕ M ) = (sin x, ϕ M) Let us prove that from the system of equations the integral identity follows for any v V h. Suppose that the representation of v has the form: then v = η ϕ η M ϕ M, (u h, v ) = (u h, η ϕ η M ϕ M) = η (u h, ϕ ) η M (u h, ϕ M) = = η (sin x, ϕ )+...+η M (sin x, ϕ M ) = (sin x, η ϕ +...+η M ϕ M ) = (sin x, v). And we complete the proof. Now let us consider the representation of an unknown function u h = ξ ϕ ξ M ϕ M (3) and substitute it in the system (3). We get (ξ ϕ ξ Mϕ M, ϕ ) = (sin x, ϕ ) (ξ ϕ ξ Mϕ M, ϕ M ) = (sin x, ϕ M) or ξ (ϕ, ϕ ) ξ M(ϕ M, ϕ ) = (sin x, ϕ ) ξ (ϕ, ϕ M) ξ M (ϕ M, ϕ M) = (sin x, ϕ M ). (4) 9

10 Finally the variational problem was reformulated in the form: Find unknown vector ξ, which satisfies the problem Aξ = b, Here the stiffness matrix A is the matrix of system (4) and the load vector b is the vector of the right-hand-sides of system (4). b) Formulate the problem (M h ), which corresponds to problem (D) in terms of stiffness matrix, load vector and coefficients of unknown function. Solution. From the representation we have v = η ϕ η M ϕ M, a(v, v) = a(η ϕ η M ϕ M, η ϕ η M ϕ M ) = = η a(ϕ, ϕ )η + η a(ϕ, ϕ 2 )η η M a(ϕ M, ϕ M )η M = η Aη, L(v) = (sin x, η ϕ η M ϕ M ) = b η, where the dot denotes the usual scalar (inner) product in IR M : ζ η = ζ η ζ M η M. Minimization problem may be formulated as: Find unknown vector ξ IR M, such that [ ] ξ Aξ b ξ = min 2 η IR M 2 η Aη b η.

### Høgskolen i Narvik Sivilingeniørutdanningen

Høgskolen i Narvik Sivilingeniørutdanningen Eksamen i Faget STE6237 ELEMENTMETODEN Klassen: 4.ID 4.IT Dato: 8.8.25 Tid: Kl. 9. 2. Tillatte hjelpemidler under eksamen: Kalkulator. Bok Numerical solution

### Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver

Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point

### Høgskolen i Narvik Sivilingeniørutdanningen

Høgskolen i Narvik Sivilingeniørutdanningen Eksamen i Faget STE66 ELASTISITETSTEORI Klasse: 4.ID Dato: 7.0.009 Tid: Kl. 09.00 1.00 Tillatte hjelpemidler under eksamen: Kalkulator Kopi av Boken Mechanics

### Numerical Solutions to Differential Equations

Numerical Solutions to Differential Equations Lecture Notes The Finite Element Method #2 Peter Blomgren, blomgren.peter@gmail.com Department of Mathematics and Statistics Dynamical Systems Group Computational

### Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### 1 Inner Products and Norms on Real Vector Spaces

Math 373: Principles Techniques of Applied Mathematics Spring 29 The 2 Inner Product 1 Inner Products Norms on Real Vector Spaces Recall that an inner product on a real vector space V is a function from

### The finite-element method

The finite-element method Version April 19, 21 [1, 2] 1 One-dimensional problems Let us apply the finite element method to the one-dimensional fin equation d 2 T dx 2 T = (1) with the boundary conditions

### 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

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### 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

### 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

### UNIVERSITETET I OSLO

NIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet Examination in: Trial exam Partial differential equations and Sobolev spaces I. Day of examination: November 18. 2009. Examination hours:

### Notes on Application of Finite Element Method to the Solution of the Poisson Equation

Notes on Application of Finite Element Method to the Solution of the Poisson Equation Consider the arbitrary two dimensional domain, shown in Figure 1. The Poisson equation for this domain can be written

### 7 - Linear Transformations

7 - Linear Transformations Mathematics has as its objects of study sets with various structures. These sets include sets of numbers (such as the integers, rationals, reals, and complexes) whose structure

### The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

### Notes on weak convergence (MAT Spring 2006)

Notes on weak convergence (MAT4380 - Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p

### Fourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) +

Fourier Series A Fourier series is an infinite series of the form a b n cos(nωx) c n sin(nωx). Virtually any periodic function that arises in applications can be represented as the sum of a Fourier series.

### Introduction to the Finite Element Method (FEM)

Introduction to the Finite Element Method (FEM) ecture First and Second Order One Dimensional Shape Functions Dr. J. Dean Discretisation Consider the temperature distribution along the one-dimensional

### 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

### Module 1 : Conduction. Lecture 5 : 1D conduction example problems. 2D conduction

Module 1 : Conduction Lecture 5 : 1D conduction example problems. 2D conduction Objectives In this class: An example of optimization for insulation thickness is solved. The 1D conduction is considered

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

### Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)

ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process

### 3.5 Spline interpolation

3.5 Spline interpolation Given a tabulated function f k = f(x k ), k = 0,... N, a spline is a polynomial between each pair of tabulated points, but one whose coefficients are determined slightly non-locally.

### DERIVATIVES AS MATRICES; CHAIN RULE

DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we

### 3.7 Non-autonomous linear systems of ODE. General theory

3.7 Non-autonomous linear systems of ODE. General theory Now I will study the ODE in the form ẋ = A(t)x + g(t), x(t) R k, A, g C(I), (3.1) where now the matrix A is time dependent and continuous on some

### (4.8) Solving Systems of Linear DEs by Elimination

INTRODUCTION: (4.8) Solving Systems of Linear DEs by Elimination Simultaneous dinary differential equations involve two me equations that contain derivatives of two me dependent variables the unknown functions

### Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 28

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 28 In the earlier lectures, we have seen formulation for 3 node linear triangular

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### Some Basic Properties of Vectors in n

These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation

### Inner product. Definition of inner product

Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product

### Chapter 20. Vector Spaces and Bases

Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit

### LS.6 Solution Matrices

LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

### 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

### 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

### Math 113 Homework 3. October 16, 2013

Math 113 Homework 3 October 16, 2013 This homework is due Thursday October 17th at the start of class. Remember to write clearly, and justify your solutions. Please make sure to put your name on the first

### Review: Vector space

Math 2F Linear Algebra Lecture 13 1 Basis and dimensions Slide 1 Review: Subspace of a vector space. (Sec. 4.1) Linear combinations, l.d., l.i. vectors. (Sec. 4.3) Dimension and Base of a vector space.

### 5 Indefinite integral

5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

### Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 01

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 01 Welcome to the series of lectures, on finite element analysis. Before I start,

### Høgskolen i Narvik- Sivilingeniørutdanningen. I FAGET STE6290 Materialvalg i Produktutforming 7.5 stp.

Høgskolen i Narvik- Sivilingeniørutdanningen EKSAMEN I FAGET STE6290 Materialvalg i Produktutforming 7.5 stp. (Material selection in Product Design) KLASSE: 4 klasse Ingeniørdesign (4ID) DATO: 6. mars

### Fourier Series Chapter 3 of Coleman

Fourier Series Chapter 3 of Coleman Dr. Doreen De eon Math 18, Spring 14 1 Introduction Section 3.1 of Coleman The Fourier series takes its name from Joseph Fourier (1768-183), who made important contributions

### Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:

Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1

### since by using a computer we are limited to the use of elementary arithmetic operations

> 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations

### We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

### An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig

An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig Abstract: This paper aims to give students who have not yet taken a course in partial

### 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

### Galerkin Approximations and Finite Element Methods

Galerkin Approximations and Finite Element Methods Ricardo G. Durán 1 1 Departamento de Matemática, Facultad de Ciencias Exactas, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina. Chapter 1 Galerkin

### The Method of Lagrange Multipliers

The Method of Lagrange Multipliers S. Sawyer October 25, 2002 1. Lagrange s Theorem. Suppose that we want to maximize (or imize a function of n variables f(x = f(x 1, x 2,..., x n for x = (x 1, x 2,...,

### Math 124A October 06, We then use the chain rule to compute the terms of the equation (1) in these new variables.

Math 124A October 06, 2011 Viktor Grigoryan 5 Classification of second order linear PDEs Last time we derived the wave and heat equations from physical principles. We also saw that Laplace s equation describes

### LINE INTEGRALS OF VECTOR FUNCTIONS: GREEN S THEOREM. Contents. 2. Green s Theorem 3

LINE INTEGRALS OF VETOR FUNTIONS: GREEN S THEOREM ontents 1. A differential criterion for conservative vector fields 1 2. Green s Theorem 3 1. A differential criterion for conservative vector fields We

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Algebra and Linear Algebra

Vectors Coordinate frames 2D implicit curves 2D parametric curves 3D surfaces Algebra and Linear Algebra Algebra: numbers and operations on numbers 2 + 3 = 5 3 7 = 21 Linear algebra: tuples, triples,...

### Orthogonal Matrices. u v = u v cos(θ) T (u) + T (v) = T (u + v). It s even easier to. If u and v are nonzero vectors then

Part 2. 1 Part 2. Orthogonal Matrices If u and v are nonzero vectors then u v = u v cos(θ) is 0 if and only if cos(θ) = 0, i.e., θ = 90. Hence, we say that two vectors u and v are perpendicular or orthogonal

### MATH301 Real Analysis Tutorial Note #3

MATH301 Real Analysis Tutorial Note #3 More Differentiation in Vector-valued function: Last time, we learn how to check the differentiability of a given vector-valued function. Recall a function F: is

### A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### 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,

### 1 Gaussian Elimination

Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

### Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem)

Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem) In order to understand the details of the Fingerprinting Theorem on fingerprints of different texts from Chapter 19 of the

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### 22 Matrix exponent. Equal eigenvalues

22 Matrix exponent. Equal eigenvalues 22. Matrix exponent Consider a first order differential equation of the form y = ay, a R, with the initial condition y) = y. Of course, we know that the solution to

### Quick Reference Guide to Linear Algebra in Quantum Mechanics

Quick Reference Guide to Linear Algebra in Quantum Mechanics Scott N. Walck September 2, 2014 Contents 1 Complex Numbers 2 1.1 Introduction............................ 2 1.2 Real Numbers...........................

### THEORY OF SIMPLEX METHOD

Chapter THEORY OF SIMPLEX METHOD Mathematical Programming Problems A mathematical programming problem is an optimization problem of finding the values of the unknown variables x, x,, x n that maximize

### CHAPTER 2 FOURIER SERIES

CHAPTER 2 FOURIER SERIES PERIODIC FUNCTIONS A function is said to have a period T if for all x,, where T is a positive constant. The least value of T>0 is called the period of. EXAMPLES We know that =

### 4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

### ( ) 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

### 3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes

Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same

### 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

### Lecture 10. Finite difference and finite element methods. Option pricing Sensitivity analysis Numerical examples

Finite difference and finite element methods Lecture 10 Sensitivities and Greeks Key task in financial engineering: fast and accurate calculation of sensitivities of market models with respect to model

### MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,

MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call

### MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### 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

### Mechanics 1: Conservation of Energy and Momentum

Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation

### Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### Linear Least Squares

Linear Least Squares Suppose we are given a set of data points {(x i,f i )}, i = 1,...,n. These could be measurements from an experiment or obtained simply by evaluating a function at some points. One

### x 2 would be a solution to d y

CATHOLIC JUNIOR COLLEGE H MATHEMATICS JC PRELIMINARY EXAMINATION PAPER I 0 System of Linear Equations Assessment Objectives Solution Feedback To use a system of linear c equations to model and solve y

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

### Chapter 4 Parabolic Equations

161 Chapter 4 Parabolic Equations Partial differential equations occur in abundance in a variety of areas from engineering, mathematical biology and physics. In this chapter we will concentrate upon the

### 5.3 The Cross Product in R 3

53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

### THREE DIMENSIONAL GEOMETRY

Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

### CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

### Fourier Series and Sturm-Liouville Eigenvalue Problems

Fourier Series and Sturm-Liouville Eigenvalue Problems 2009 Outline Functions Fourier Series Representation Half-range Expansion Convergence of Fourier Series Parseval s Theorem and Mean Square Error Complex

### (Positive) Rational Numbers

(Positive) Rational Numbers Definition Definition 1 Consider all ordered pairs (, ) with, N Define the equivalance (, ) (y 1, ) = y 1 (1) Theorem 2 is indeed an equivalence relation, that is 1 (, ) (,

### 17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):

### Homework One Solutions. Keith Fratus

Homework One Solutions Keith Fratus June 8, 011 1 Problem One 1.1 Part a In this problem, we ll assume the fact that the sum of two complex numbers is another complex number, and also that the product

### Solutions to Linear First Order ODE s

First Order Linear Equations In the previous session we learned that a first order linear inhomogeneous ODE for the unknown function x = x(t), has the standard form x + p(t)x = q(t) () (To be precise we

### LECTURE NOTES: FINITE ELEMENT METHOD

LECTURE NOTES: FINITE ELEMENT METHOD AXEL MÅLQVIST. Motivation The finite element method has two main strengths... Geometry. Very complex geometries can be used. This is probably the main reason why finite

### Scalar Valued Functions of Several Variables; the Gradient Vector

Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector) valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x) = φ(x 1,

### CHANGE OF VARIABLES AND THE JACOBIAN. Contents 1. Change of variables: the Jacobian 1

CHANGE OF VARIABLES AND THE JACOBIAN Contents 1. Change of variables: the Jacobian 1 1. Change of variables: the Jacobian So far, we have seen three examples of situations where we change variables to

### 1 Introduction to Matrices

1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

### 6. ISOMETRIES isometry central isometry translation Theorem 1: Proof:

6. ISOMETRIES 6.1. Isometries Fundamental to the theory of symmetry are the concepts of distance and angle. So we work within R n, considered as an inner-product space. This is the usual n- dimensional

### 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