# Scalar Valued Functions of Several Variables; the Gradient Vector

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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, x 2,..., x n, X D, where the domain D is a region in R n. By this we mean that D is an open, connected set. The term open means that each point X 0 D is the center of a ball N(X 0, r = { X X X0 < r }, for some r > 0, which still lies in D. The term connected, as we will use it here, means that any two points in D can be connected by a curve lying in D. The partial derivatives of the function φ(x at a point X 0 D are the numbers, which we denote by (X 0, k = 1, 2,..., n, obtained by differentiating the function φ(x with respect to the variable x k at the point X 0, treating the other variables, x 1,..., x k 1, x k+1,..., x n, as constant quantities. Taking the components of X 0 to be x 0,k, k = 1, 2,..., n, these partial derivatives exist just in case the corresponding limits of difference quotients φ(x 0,1,..., x 0,k 1, x k, x 0,k+1,..., x 0,n φ(x 0,1,..., x 0,k 1, x 0,k, x 0,k+1,..., x 0,n lim x k x 0,k x k x 0,k exist; the limit values are the partial derivatives we have indicated. As we allow the point X 0 to range throughout the region D the relationship thus engendered between X 0 and the partial derivative (X 0 defines a function; replacing X 0 by the symbol X, we denote this function by (X. It is also common to see this function written as φ xk (X = φ xk (x 1, x 2,..., x n. If each of these functions is continuous as a function of X D then we say that φ(x is continuously differentiable in the region D. In the work to follow we will assume continuous differentiability unless we specifically indicate otherwise. If we combine the partial derivatives (X 0, k = 1, 2,..., n into an n-dimensional vector, in the obvious order, we obtain the gradient (vector of the function φ(x at 1

2 the point X 0. A commonly used notation for the gradient at X 0 is φ(x 0. Again letting X 0 range throughout D, and replacing X 0 by X we obtain the vector function ( φ(x = φ(x 1, x 2,..., x n = (X, (X,..., (X. x 1 x 2 x n In R 3 we can write φ(x = = x ( x (x, y, z, y (x, y, zi + y (x, y, z, (x, y, z z (x, y, zj + (x, y, zk. z In this way we obtain a vector field in the region D, since the dimension of φ(x is n, the same as the dimension of the independent vector variable X. Example 1 If, in R 2, we define φ(x, y = x 2 y + y3 3, then the corresponding gradient field is ( φ(x, y = x (x2 y + y3 3, y (x2 y + y3 3 = ( 2xy, x 2 + y 2. The gradient vector at the point (1, 2, for example, is (4, 5. A Notational Convention Vectors can be written in row form: (x 1, x 2,..., x n, or in column form: x 1 x 2.. x n These vectors are said to be transposes of each other; if the row vector is designated as X, then the corresponding column vector is designated as X, and vice versa. If X and Y are n-dimensional column and row vectors, respectively, it is common to write x 1 X Y = Y (X = (y 1 y 2 x 2... y n. = n y k x k. x n 2

3 If W is an n-dimensional row vector then W X n w k x k. Gradients are usually written as row vectors; thus the preceding implies that φ(x 0 X n (X 0 x k. We will use the notation repeatedly in the sequel. First Order Linear Approximation to a Scalar Function If φ(x is a continuously differentiable function of X in the region D which contains the point X 0 the gradient φ(x 0 can be used to approximate values of the function φ(x at points X D lying close to X 0. The formula for this approximation relationship is φ(x φ(x 0 + φ(x 0 (X X 0 = n (X 0 (x k x 0,k. The right hand side defines a linear function of the vector variable X, which we call the linear approximation to φ(x at the point X 0. We will denote this linear function by L φ,x0 (X. Figure 1 3

4 This relationship can be illustrated for n = 2, φ = φ(x, y by noting that in this case the graph of z = L φ,x0,y 0 (x, y is a plane in R 3 which is tangent to the graph of z = φ(x, y at the point (x 0, y 0, φ(x 0, y 0. Example 2 Let w = x 2 + y 3 + z 4. Taking the base point X 0 = (x 0, y 0, z 0 = (1, 1, 1 we have w 0 φ(1, 1, 1 = 3. Now φ(x, y, z = (2x 3y 4z ; φ(1, 1, 1 = ( Accordingly, the linear approximation to φ(x, y, z as given by the above formula is x (x 1 + 3(y 1 + 4(z 1 = w = 3 + ( y 1 =. z x + 3y + 4z The linear approximation can be used to estimate the value of φ at points X near the base point X 0 (with some degree of error corresponding to the term o ( X X 0. Thus, in the example just studied, if we take (x, y, z = (1.1,.9, 1.05, the linear approximation gives the value L φ,(1,1,1 (1.1,.9, 1.05 = 6 + 2( (.9 + 4(1.05 = 3.1, whereas the actual value is φ(1.1,.9, 1.05 = ( ( ( = Basis for the Approximation; Error We want to see why the linear approximation, involving the gradient vector, works as it does, and to obtain a bound on the error incurred in use of this approximation. For clarity and brevity we carry out the calculations for the case n = 2; the argument in a larger number of variables is essentially identical. Let φ(x, y be continuously differentiable in a region D, let X 0 = (x 0, y 0 be a point in D and let N(X 0, r be a disc of positive radius, r, centered at X 0, which still lies in D. Let X = (x, y be a point in N(X 0, r. We consider the difference φ(x φ(x 0 = φ(x, y φ(x 0, y 0 = φ(x, y φ(x, y 0 + φ(x, y 0 φ(x 0, y 0. 4

5 Applying the Mean Value Theorem twice, we have φ(x, y φ(x, y 0 + φ(x, y 0 φ(x 0, y 0 = y (x, η(y y 0 + x (ξ, y 0(x x 0, where ξ lies between x 0 and x and η lies between y 0 and y. Then, clearly, + φ(x, y φ(x 0, y 0 = y (x 0, y 0 (y y 0 + x (x 0, y 0 (x x 0 ( ( (x, η y y (x 0, y 0 (y y 0 + x (ξ, y 0 x (x 0, y 0 (x x 0 = φ(x 0, y 0 (X X 0 + γ(x 0, X(X X 0, where, in the error term γ(x 0, X(X X 0, γ(x 0, X = ( (x, η (x y y 0, y 0 (ξ, y x 0 (x x 0, y 0. Applying the Schwarz inequality to the error term we have γ(x 0, X(X X 0 γ(x 0, X X X 0. Because φ(x is continuously differentiable, lim X X0 γ(x 0, X = lim ( X X y (x, η y (x 0, y 0 0 x (ξ, y 0 x (x 0, y 0 = 0. This means that γ(x 0, X(X X 0 γ(x 0, X X X 0 has the property γ(x 0, X(X X 0 lim X X 0 X X 0 a relationship which we express by saying that = 0, φ(x = φ(x 0 + φ(x 0 (X X 0 + o( X X 0 ; i.e., the error in the approximation tends to zero, as X X 0 0, more rapidly than any multiple of X X 0. It should be noted that the error estimate just obtained depends critically on the continuity of the partial derivatives of the function φ(x. Without that property one can show, with appropriate examples, that such an estimate need not hold. The approximation relationship holds in exactly the same way in R n, for a general positive integer n, as we have shown it to hold for n = 2. 5

6 Proposition (Chain Rule Suppose X(t is a continuously differentiable curve in R n and φ(x is a continuously differentiable function of X D R n. Then d dt φ(x(t = φ(x(tx (t. Fix a value of t, say t 0. Then X(t = X(t 0 + X (t 0 (t t 0 + o( t t 0 as t t 0. Using the formula for the first order approximation to φ(x at X = X(t 0 we find that φ(x(t = φ(x(t 0 + φ(x(t 0 (X(t X(t 0 + o( X(t X(t 0 ( = φ(x(t 0 + φ(x(t 0 X(t 0 + X (t 0 (t t 0 + o( t t 0 X(t 0 +o ( φ(x(t 0 (X (t 0 (t t 0 + o( t t 0 = φ(x(t 0 + φ(x(t 0 Dividing by t t 0 we then have ( X (t 0 (t t 0 + o( t t 0. φ(x(t φ(x(t 0 t t 0 = φ(x(t 0 X (t 0 + o( t t 0 t t 0. Letting t t 0 the difference quotient on the left approaches d dt φ(x(t t=t0 while the expression at the right approaches φ(x(t 0 X (t 0. Thus dφ dt (X(t = φ(x(t 0 X (t 0. t=t0 Since t 0 could be any value of t, the result follows. An important special case occurs when we take X(t to be the straight line We then obtain, since dx dt U, X(t = X 0 + t U, U = 1. dφ dt (X 0 = dφ dt (X(0 = φ(x 0U dx du (X 0. 6

7 This is called the directional derivative of φ(x in the direction U at the point X 0. It represents the slope of the graph of φ(x in the direction corresponding to the unit vector U. Using the Schwarz inequality again we obtain dφ dt (X 0 = φ(x 0 U φ(x 0 U with the inequality holding strictly unless U is collinear with φ(x 0. The two unit vectors collinear with φ(x 0 are U ± = ± φ(x 0 φ(x 0, provided φ(x 0 is not the zero (column vector. For those the corresponding directional derivatives are φ(x 0 U ± = ± φ(x 0 φ(x 0 φ(x 0 = ± φ(x 0. The gradient vectors U +, U thus correspond to the extreme values of the directional derivative U (X 0; the uphill direction U + yields its maximum and the downhill direction U yields its minimum, those maximum and minimum values being φ(x 0 and φ(x 0, respectively. Suggested Exercises 1. Find the gradients, as vector functions of X, for φ(x = φ(x, y = sin(x 2 +y 2 +2x 3y; ψ(x = ψ(x, y, z = x 2 + 2xy 2 +y 4 z Find the linear approximation to ψ(x, y, z, as defined in 2.,at the point X = 1 1. Use the linear approximation to estimate the values of φ at the points and Where is the approximation most effective? Why? Let φ(x, y be as defined in 1. Find all unit vectors U such that ( 1, 2 = 0. U 4. Let X(t be the curve described by X(t = t + 1 t 3 t 2 t and let ψ(x be as defined in 1. Compute d ψ(x(t as a function of t and give its value when t = 1. dt 7

### 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)

### 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

### Solutions for Review Problems

olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

### 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

### 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

### 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

### 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

### Taylor and Maclaurin Series

Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

### vector calculus 2 Learning outcomes

29 ontents vector calculus 2 1. Line integrals involving vectors 2. Surface and volume integrals 3. Integral vector theorems Learning outcomes In this Workbook you will learn how to integrate functions

### JUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson

JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3

### FINAL EXAM SOLUTIONS Math 21a, Spring 03

INAL EXAM SOLUIONS Math 21a, Spring 3 Name: Start by printing your name in the above box and check your section in the box to the left. MW1 Ken Chung MW1 Weiyang Qiu MW11 Oliver Knill h1 Mark Lucianovic

### the points are called control points approximating curve

Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.

### AP CALCULUS AB 2009 SCORING GUIDELINES

AP CALCULUS AB 2009 SCORING GUIDELINES Question 5 x 2 5 8 f ( x ) 1 4 2 6 Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

### INTERPOLATION. Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y).

INTERPOLATION Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). As an example, consider defining and x 0 =0, x 1 = π 4, x

### 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,

### This makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5

1. (Line integrals Using parametrization. Two types and the flux integral) Formulas: ds = x (t) dt, d x = x (t)dt and d x = T ds since T = x (t)/ x (t). Another one is Nds = T ds ẑ = (dx, dy) ẑ = (dy,

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Parametric Curves. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015

Parametric Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 8, 2015 1 Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final point

### SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve

SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives

### Vector Spaces; the Space R n

Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which

### SF2940: Probability theory Lecture 8: Multivariate Normal Distribution

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,

### Systems with Persistent Memory: the Observation Inequality Problems and Solutions

Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +

### Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

### ( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those

1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make

### New Higher-Proposed Order-Combined Approach. Block 1. Lines 1.1 App. Vectors 1.4 EF. Quadratics 1.1 RC. Polynomials 1.1 RC

New Higher-Proposed Order-Combined Approach Block 1 Lines 1.1 App Vectors 1.4 EF Quadratics 1.1 RC Polynomials 1.1 RC Differentiation-but not optimisation 1.3 RC Block 2 Functions and graphs 1.3 EF Logs

### Level Set Framework, Signed Distance Function, and Various Tools

Level Set Framework Geometry and Calculus Tools Level Set Framework,, and Various Tools Spencer Department of Mathematics Brigham Young University Image Processing Seminar (Week 3), 2010 Level Set Framework

### 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

### 1 3 4 = 8i + 20j 13k. x + w. y + w

) Find the point of intersection of the lines x = t +, y = 3t + 4, z = 4t + 5, and x = 6s + 3, y = 5s +, z = 4s + 9, and then find the plane containing these two lines. Solution. Solve the system of equations

### Section 1.5 Linear Models

Section 1.5 Linear Models Some real-life problems can be modeled using linear equations. Now that we know how to find the slope of a line, the equation of a line, and the point of intersection of two lines,

### Tutorial on Using Excel Solver to Analyze Spin-Lattice Relaxation Time Data

Tutorial on Using Excel Solver to Analyze Spin-Lattice Relaxation Time Data In the measurement of the Spin-Lattice Relaxation time T 1, a 180 o pulse is followed after a delay time of t with a 90 o pulse,

### 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.

### Equations Involving Lines and Planes Standard equations for lines in space

Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity

### General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1

A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions

### Surface Normals and Tangent Planes

Surface Normals and Tangent Planes Normal and Tangent Planes to Level Surfaces Because the equation of a plane requires a point and a normal vector to the plane, nding the equation of a tangent plane to

### Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations

Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate

### Review of Vector Analysis in Cartesian Coordinates

R. evicky, CBE 6333 Review of Vector Analysis in Cartesian Coordinates Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers.

### MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI

### Solutions to Homework 5

Solutions to Homework 5 1. Let z = f(x, y) be a twice continously differentiable function of x and y. Let x = r cos θ and y = r sin θ be the equations which transform polar coordinates into rectangular

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve

QUALITATIVE THEORY OF DYAMICAL SYSTEMS 2, 61 66 (2001) ARTICLE O. 11 Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve Alexei Grigoriev Department of Mathematics, The

### L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

### The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

### Electrical Engineering 103 Applied Numerical Computing

UCLA Fall Quarter 2011-12 Electrical Engineering 103 Applied Numerical Computing Professor L Vandenberghe Notes written in collaboration with S Boyd (Stanford Univ) Contents I Matrix theory 1 1 Vectors

### Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.

Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve

### 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

### 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

### 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

### Introduction to Partial Differential Equations. John Douglas Moore

Introduction to Partial Differential Equations John Douglas Moore May 2, 2003 Preface Partial differential equations are often used to construct models of the most basic theories underlying physics and

### Section 9.5: Equations of Lines and Planes

Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that

### Lecture 3: Derivatives and extremes of functions

Lecture 3: Derivatives and extremes of functions Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2011 Lejla Batina Version: spring 2011 Wiskunde 1 1 / 16

### Numerisches Rechnen. (für Informatiker) M. Grepl J. Berger & J.T. Frings. Institut für Geometrie und Praktische Mathematik RWTH Aachen

(für Informatiker) M. Grepl J. Berger & J.T. Frings Institut für Geometrie und Praktische Mathematik RWTH Aachen Wintersemester 2010/11 Problem Statement Unconstrained Optimality Conditions Constrained

### 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.

### Figure 2.1: Center of mass of four points.

Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would

### 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

### Calculus. Contents. Paul Sutcliffe. Office: CM212a.

Calculus Paul Sutcliffe Office: CM212a. www.maths.dur.ac.uk/~dma0pms/calc/calc.html Books One and several variables calculus, Salas, Hille & Etgen. Calculus, Spivak. Mathematical methods in the physical

### 2.2 Creaseness operator

2.2. Creaseness operator 31 2.2 Creaseness operator Antonio López, a member of our group, has studied for his PhD dissertation the differential operators described in this section [72]. He has compared

### 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

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### Given three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B);

1.1.4. Prouct of three vectors. Given three vectors A, B, anc. We list three proucts with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); a 1 a 2 a 3 (A B) C = b 1 b 2 b 3 c 1 c 2 c 3 where the

### Vector Spaces. Chapter 2. 2.1 R 2 through R n

Chapter 2 Vector Spaces One of my favorite dictionaries (the one from Oxford) defines a vector as A quantity having direction as well as magnitude, denoted by a line drawn from its original to its final

### An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate.

Chapter 10 Series and Approximations An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate 1 0 e x2 dx, you could set

### 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

### Slope and Rate of Change

Chapter 1 Slope and Rate of Change Chapter Summary and Goal This chapter will start with a discussion of slopes and the tangent line. This will rapidly lead to heuristic developments of limits and the

### 3.2 The Factor Theorem and The Remainder Theorem

3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial

### H/wk 13, Solutions to selected problems

H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.

### Section 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### Macroeconomics Lecture 1: The Solow Growth Model

Macroeconomics Lecture 1: The Solow Growth Model Richard G. Pierse 1 Introduction One of the most important long-run issues in macroeconomics is understanding growth. Why do economies grow and what determines

### 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,

### 4. Expanding dynamical systems

4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,

### Lecture 4: BK inequality 27th August and 6th September, 2007

CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound

### Discrete Convolution and the Discrete Fourier Transform

Discrete Convolution and the Discrete Fourier Transform Discrete Convolution First of all we need to introduce what we might call the wraparound convention Because the complex numbers w j e i πj N have

### The Relation between Two Present Value Formulae

James Ciecka, Gary Skoog, and Gerald Martin. 009. The Relation between Two Present Value Formulae. Journal of Legal Economics 15(): pp. 61-74. The Relation between Two Present Value Formulae James E. Ciecka,

### Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 25 Notes

Jim Lambers MAT 169 Fall Semester 009-10 Lecture 5 Notes These notes correspond to Section 10.5 in the text. Equations of Lines A line can be viewed, conceptually, as the set of all points in space that

### 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

### Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

### Introduction to Differential Calculus. Christopher Thomas

Mathematics Learning Centre Introduction to Differential Calculus Christopher Thomas c 1997 University of Sydney Acknowledgements Some parts of this booklet appeared in a similar form in the booklet Review

### FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

### 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

### K 1 < K 2 = P (K 1 ) P (K 2 ) (6) This holds for both American and European Options.

Slope and Convexity Restrictions and How to implement Arbitrage Opportunities 1 These notes will show how to implement arbitrage opportunities when either the slope or the convexity restriction is violated.

### 4. Complex integration: Cauchy integral theorem and Cauchy integral formulas. Definite integral of a complex-valued function of a real variable

4. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complex-valued function of a real variable Consider a complex valued function f(t) of a real variable

### Critical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.

Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### 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

### 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

### 4.2. LINE INTEGRALS 1. 2 2 ; z = t. ; y = sin

4.2. LINE INTEGRALS 1 4.2 Line Integrals MATH 294 FALL 1982 FINAL # 7 294FA82FQ7.tex 4.2.1 Consider the curve given parametrically by x = cos t t ; y = sin 2 2 ; z = t a) Determine the work done by the

### 2.2. Instantaneous Velocity

2.2. Instantaneous Velocity toc Assuming that your are not familiar with the technical aspects of this section, when you think about it, your knowledge of velocity is limited. In terms of your own mathematical

### About the inverse football pool problem for 9 games 1

Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

### Convex analysis and profit/cost/support functions

CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m

### SOLUTIONS. Would you like to know the solutions of some of the exercises?? Here you are

SOLUTIONS Would you like to know the solutions of some of the exercises?? Here you are Your first function : remember Introduction: Text: A function is a relationship between two sets by which we assign

### The Basics of Interest Theory

Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest

### Some Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)

Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals,

### 1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.

.(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3

### 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