# To give it a definition, an implicit function of x and y is simply any relationship that takes the form:

Save this PDF as:

Size: px
Start display at page:

Download "To give it a definition, an implicit function of x and y is simply any relationship that takes the form:"

## Transcription

1 2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to think of this theorem when you want to figure out how a change in variable x affects variable y First a reminder of what an implicit function looks like A non-implicit function (an explicit function?) is the kind of thing you re accustomed to, which looks like: y = f(x) and you call y the dependent variable and x the independent variable But you could also express this same relationship as a zero of the implicit function: g(x, y) = y f(x The pairs of x and y which satisfy the first relationship will also satisfy the second relationship The reason for calling it an implicit function is that it doesn t say outright that y depends on x but it is a function: as you vary x, you have to vary y as well, in order to maintain the equals zero relationship To give it a definition, an implicit function of x and y is simply any relationship that takes the form: g(x, y Any explicit function can be changed into an implicit function using the trick above, just setting g(x, y) = y f(x In theory, any implicit function could be converted into an explicit function by solving for y in terms of x In practice, this may be rather challenging, though Consider: ln(x + y)+ xy 12 = 0 Try as hard as you like, you ll never be able to isolate either of the variables This is one motivation for working with implicit functions Another motivation is that we often work with general functions, rather than a particular functional form For instance, once can generally state the FOC from a typical utility maximization problem in this form: u( v x ) λp l = 0 It would be desirable to talk about properties of demands without assuming that the utility function is Cobb-Douglas, CES, or anything in particular a general form would encompass all these cases Fall 2001 math for economic theory, page 6

2 The final reason to learn how to work with implicit functions is that implicit function naturally arise in economics Every time we do a constrained optimization problem, we end up with some condition set equal to zero This is already an implicit function Why try to solve it for one variable in terms of the others, when we don t need to? These are the sorts of things that we will be asking from the implicit function theorem: Example 211: u w 1 s The first order condition from a utility maximization problem is ( ) + (1 + r) u ( w 2 + (1 + r)s Find ds dr Example 212: The first order conditions from a utility maximization problem are: α αx 1 1 x 1 α 2 λp 1 = 0 (1 α)x α 1 x α 2 λp 2 = 0 Find all the partial derivatives of the demand functions, x i p j 22 The implicit function theorem (two variable case) When we have an implicit function of the form g(x, y, x, y R 1, the implicit function theorem says that we can figure out dy dx quite easily 1 Provided that we have some continuity and a non-zero denominator, this derivative is: g(x, y dy dx = x y Remember that for dy dx, the denominator of the right-hand side is y In other words, whatever variable is on top on the left is the derivative which is on bottom on the right One trick to remember this is to pretend that you can cancel out the whatever terms, so that you get: x y = g / x g / y = 1 x 1 y = y x This is not how the implicit function theorem works, though It s just a useful trick for checking what goes where 1 At least, you can figure it out quite easily provided that you know which derivative you re trying to find these variables aren t going to be called x and y in the problems you encounter, they ll be s and r and p and h and w Half the time, the question won t ask, Find s r as in Example 211 instead it ll be worded, Find how much savings changes when the interest rate changes It ll be up to you to figure out what that means Fall 2001 math for economic theory, page 7

3 So how does the implicit function theorem work? Again, an implicit function is like a normal function in that y must change as x changes, in order to maintain the relationship You can think of y(x) The function is really just a function of one independent variable, like this: ( g x, y(x) There s a y in there, but not really the value of y is dictated by x By totally differentiating this function with respect to x, we get: d dx [ g( x, y(x) )] = d [ dx 0 ] x + y y (x) = 0 dy dx = y (x) = x y And thus, we get the implicit function theorem by doing nothing more than treating y as a function of x and totally differentiating An alternative way of deriving this result is to just take the function g(x, y), and write its change in differential form Since we know that 0 is unchanging, this form is: dy + y x dx = 0 dy = y x dx dy x = dx y The derivation of the implicit function theorem is quick and simple, and it might be work remembering in case you can t remember the theorem itself 23 ultivariate versions of the implicit function theorem When y is an implicit function of many variables Now we re talking about implicit functions that look like: g, y It turns out that nothing really changes for these functions If we write out the differential form of this function, we get: dx 1 + dx 2 +K + dx n + x 2 x n y dy = 0 Fall 2001 math for economic theory, page 8

4 By setting all the dx i = 0 except for one dx l, we get the partial derivative of y with respect to x l This expression turns out to be much the same as for the single-x case: y = y When there are many y variables and many x variables Let s say that we have m variables that we call y 1,K y m, and that these variables are implicitly a function of some n variables, called x 1 It will take m equations to describe the entire system of y variables, like: g 1 g 2 g m There are several ways to express the implicit function theorem in this form One is to imagine that those m implicit functions are a single vector-valued function g:r n m R m, such that: g( v x, v y v If we write this in differential form, we get that: Dx v ( x v, v y ) dx { + D v y ( v x, y v ) n dy { = 0 { m 1 m n m m m 1 (You can verify that the dimensions make sense, right?) You can sort of solve this equation to get one multivariate formulation of the theorem: dy = D v y g( x v, v y ) 1 Dx v g( v x, v y ) dx Alternatively, you could think about only the partial derivatives of y with respect to one of the x variables To find this out, you do: T,,K = Dy x i x i x i v g( v x, y v ) 1 D xi g( v x, v y ) Finally, we can use a version of Cramer s rule to solve systems of implicit functions Suppose that we re given: g 1 g 2 g m and we want to know one partial derivative, like y k this function looks like: The differential form of Fall 2001 math for economic theory, page 9

5 1 dy 1 +K + 1 dy m + 1 dx 1 + K+ 1 x n dx n = 0 m dy 1 +K + m dy m + m dx 1 +K + m x n dx n = 0 Now, the idea of a partial derivative is that only x l changes and none of the other x variables; but this can still mean that lots of y variables change around, not just y k So we set dx i = 0 for all i l (which means that we re really talking about a partial derivative at this point holding everything constant except dx l I ll change the notation to partial derivatives) This results in a system of m equations in m unknowns these unknowns are the y j : 1 +K + 1 = 1 m +K + m = m (See how it s a system of linear equations?) There are several ways that we can solve this system of equations one of them being Cramer s rule Recall that if we want to solve for the unknown variable y k on the left-hand side, we establish the m m matrix of coefficients on the unknown y j variables this turns out to be the same as the matrix of first derivatives, D y g( v x, y v ) Cramer s rule says that y k is equal to a fraction, the denominator of which is det( D y g( v x, y v )), the numerator of which is the determinant of the same matrix with the k-th column replaced by the vector on the right-hand side, or ore clearly: 1 L 1 L 1 det m y k L m L m = l L 1 L 1 det m y1 L m L m That s the formula to remember for the multivariate version of the implicit function theorem Note that the minus sign in front of the in the system of equations above has turned into a minus sign out in front of the fraction This comes because we ve changed the signs of every element in a row of the matrix in the numerator; there s a rule which says that multiplying every element of one row or column of a matrix by some scalar α means that the determinant of the matrix also changes by the same α In this case, α = 1 Fall 2001 math for economic theory, page 10

6 24 Applications and exercises Some applications of the implicit function theorem show up in micro, but most of them are in macro Here are some problems that you will face: Exercise 241: For the utility function u, x 2 ), a particular vector, x 2 ) will put you on some indifference curve u Find the slope of this curve at, x 2 ) Step 1: Figure out what derivative you want Step 2:Define an implicit function of these variables Step 3: Apply the implicit function theorem Exercise 242: The individual lives for two periods He has a utility function u(c 1 )+ βu(c 2 ) His budget constraint requires that his period 1 consumption be his period 1 endowment minus any savings, c 1 = w 1 s In the second period, his consumption will be c 2 = w 2 + (1 + r)θs + α The government has taxed savings at the rate 1 θ, and uses it to finance a lump-sum transfer of α The individual takes θ and α as given, but for the government the government s budget to balance, we must have α = (1 θ)(1 + r)s Find s θ Step 1: Set up the utility maximization problem Step 2:Find FOCs These should define an implicit function Step 3: Insert the GBC, after taking the FOC Step 4:Apply the implicit function theorem Exercise 243: People live for two periods (in overlapping generations), and the utility function is the same as before The government has implemented a pay-as-you-go social security system The way this works is that every young person pays a lump-sum tax of θ, and every old person collects a pension of α This means that c 1 = w 1 θ and c 2 = w 2 + (1 + r)s + α The size of each cohort (or generation) increases at rate n, which means that N t +1 = (1 + n)n t, where N t represents the number of people born at time t Find s θ Step 1: Set up the utility maximization problem Step 2:Figure out what the GBC is supposed to be Step 3: Find FOCs Step 4: Insert the GBC Step 5: Apply the implicit function theorem Fall 2001 math for economic theory, page 11

7 Exercise 244: The individual lives for two periods His utility function depends on consumption at time 1 and consumption at time 2, u(c 1,c 2 ) He is endowed with w 1 when young and w 2 when old He can chose savings s at t = 1, which get a gross rate of return 1 + r For each of the following utility functions, find s r a u(c 1,c 2 ) = ln c 1 + β ln c 2 b u(c 1,c 2 ) = ln c 1 + β c 2 c u(c 1,c 2 ) = (1 ρ) 1 c 1 1 ρ + β c c 1 ρ { }, and w 2 = 0 d u(c 1,c 2 ) = exp( ρc 1 )+ exp( ρc 2 ), and w 2 = 0 Step 1: Sep up the utility maximization problem Step 2:Find FOCs Step 3: Evaluate: is it easier to solve for s, or to use the IFT? Step 4:Do it If you have some extra time, you might want to try solving those problems using both methods solving for s directly, as well as using the implicit function theorem Which way is easier? What s different about the way the answers look? Can you reconcile them? Exercise 245: The individual has a utility function represented by u, x 2 ) = x α 1 α 1 x 2 Using the first order conditions, find x i p j Now solve the demand functions, and take the same derivatives Do they match up? Why or why not? Exercise 246: Same thing, except that the utility function is now ( ) 1/ρ u, x 2 ) = x 1 ρ + x 2 ρ Fall 2001 math for economic theory, page 12

### Multi-variable Calculus and Optimization

Multi-variable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 1 / 51 EC2040 Topic 3 - Multi-variable Calculus

### Constrained optimization.

ams/econ 11b supplementary notes ucsc Constrained optimization. c 2010, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values

### Lecture 4: Equality Constrained Optimization. Tianxi Wang

Lecture 4: Equality Constrained Optimization Tianxi Wang wangt@essex.ac.uk 2.1 Lagrange Multiplier Technique (a) Classical Programming max f(x 1, x 2,..., x n ) objective function where x 1, x 2,..., x

### 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

### Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

### Chapter 4 Online Appendix: The Mathematics of Utility Functions

Chapter 4 Online Appendix: The Mathematics of Utility Functions We saw in the text that utility functions and indifference curves are different ways to represent a consumer s preferences. Calculus can

### Separable First Order Differential Equations

Separable First Order Differential Equations Form of Separable Equations which take the form = gx hy or These are differential equations = gxĥy, where gx is a continuous function of x and hy is a continuously

### 1 Lecture: Integration of rational functions by decomposition

Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

### Joint Probability Distributions and Random Samples (Devore Chapter Five)

Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### Matrix Algebra and Applications

Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2 - Matrices and Matrix Algebra Reading 1 Chapters

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### 2 Integrating Both Sides

2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation

### We know a formula for and some properties of the determinant. Now we see how the determinant can be used.

Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we

### Calculus with Analytic Geometry I Exam 5-Take Home Part Due: Monday, October 3, 2011; 12PM

NAME: Calculus with Analytic Geometry I Exam 5-Take Home Part Due: Monday, October 3, 2011; 12PM INSTRUCTIONS. As usual, show work where appropriate. As usual, use equal signs properly, write in full sentences,

### 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

### Methods for Finding Bases

Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

### Integrals of Rational Functions

Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t

### Deriving Demand Functions - Examples 1

Deriving Demand Functions - Examples 1 What follows are some examples of different preference relations and their respective demand functions. In all the following examples, assume we have two goods x

### Linear and quadratic Taylor polynomials for functions of several variables.

ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is

### 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

### 2.3 Solving Equations Containing Fractions and Decimals

2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### 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

### Homework #2 Solutions

MAT Spring Problems Section.:, 8,, 4, 8 Section.5:,,, 4,, 6 Extra Problem # Homework # Solutions... Sketch likely solution curves through the given slope field for dy dx = x + y...8. Sketch likely solution

### (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

### Chapter 8. Matrices II: inverses. 8.1 What is an inverse?

Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we

### Name: Section Registered In:

Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

### 1 Sufficient statistics

1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =

### SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA

SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the

### Linear Programming Problems

Linear Programming Problems Linear programming problems come up in many applications. In a linear programming problem, we have a function, called the objective function, which depends linearly on a number

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### Differentiation and Integration

This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

### 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

### Prot Maximization and Cost Minimization

Simon Fraser University Prof. Karaivanov Department of Economics Econ 0 COST MINIMIZATION Prot Maximization and Cost Minimization Remember that the rm's problem is maximizing prots by choosing the optimal

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

### Name: ID: Discussion Section:

Math 28 Midterm 3 Spring 2009 Name: ID: Discussion Section: This exam consists of 6 questions: 4 multiple choice questions worth 5 points each 2 hand-graded questions worth a total of 30 points. INSTRUCTIONS:

### Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

### 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

### Linear Algebra Notes

Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### 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

### Solving DEs by Separation of Variables.

Solving DEs by Separation of Variables. Introduction and procedure Separation of variables allows us to solve differential equations of the form The steps to solving such DEs are as follows: dx = gx).

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### 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

### 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

### MATH 2300 review problems for Exam 3 ANSWERS

MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test

### 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

### 3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or

### Solving Systems of Linear Equations

LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

### 1 Homogenous and Homothetic Functions

1 Homogenous and Homothetic Functions Reading: [Simon], Chapter 20, p. 483-504. 1.1 Homogenous Functions Definition 1 A real valued function f(x 1,..., x n ) is homogenous of degree k if for all t > 0

### Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,

### Linear Programming Notes VII Sensitivity Analysis

Linear Programming Notes VII Sensitivity Analysis 1 Introduction When you use a mathematical model to describe reality you must make approximations. The world is more complicated than the kinds of optimization

### 5.4 Solving Percent Problems Using the Percent Equation

5. Solving Percent Problems Using the Percent Equation In this section we will develop and use a more algebraic equation approach to solving percent equations. Recall the percent proportion from the last

### 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

### 5 Homogeneous systems

5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m

### First Order Non-Linear Equations

First Order Non-Linear Equations We will briefly consider non-linear equations. In general, these may be much more difficult to solve than linear equations, but in some cases we will still be able to solve

### 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 variable λ is a dummy variable called a Lagrange multiplier ; we only really care about the values of x, y, and z.

Math a Lagrange Multipliers Spring, 009 The method of Lagrange multipliers allows us to maximize or minimize functions with the constraint that we only consider points on a certain surface To find critical

### Change of Continuous Random Variable

Change of Continuous Random Variable All you are responsible for from this lecture is how to implement the Engineer s Way (see page 4) to compute how the probability density function changes when we make

### Envelope Theorem. Kevin Wainwright. Mar 22, 2004

Envelope Theorem Kevin Wainwright Mar 22, 2004 1 Maximum Value Functions A maximum (or minimum) value function is an objective function where the choice variables have been assigned their optimal values.

### Constrained Optimisation

CHAPTER 9 Constrained Optimisation Rational economic agents are assumed to make choices that maximise their utility or profit But their choices are usually constrained for example the consumer s choice

### 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

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### 8.1. Cramer s Rule for Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Cramer s Rule for Solving Simultaneous Linear Equations 8.1 Introduction The need to solve systems of linear equations arises frequently in engineering. The analysis of electric circuits and the control

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### Vector Math Computer Graphics Scott D. Anderson

Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about

### Insurance. Michael Peters. December 27, 2013

Insurance Michael Peters December 27, 2013 1 Introduction In this chapter, we study a very simple model of insurance using the ideas and concepts developed in the chapter on risk aversion. You may recall

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### 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

### Recognizing Types of First Order Differential Equations E. L. Lady

Recognizing Types of First Order Differential Equations E. L. Lady Every first order differential equation to be considered here can be written can be written in the form P (x, y)+q(x, y)y =0. This means

### Stanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions

Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in

### Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

### 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.

### 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field

3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important

### 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

### Sometimes it is easier to leave a number written as an exponent. For example, it is much easier to write

4.0 Exponent Property Review First let s start with a review of what exponents are. Recall that 3 means taking four 3 s and multiplying them together. So we know that 3 3 3 3 381. You might also recall

### Solution to Homework 2

Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if

### " Y. Notation and Equations for Regression Lecture 11/4. Notation:

Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

### 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

### Math 53 Worksheet Solutions- Minmax and Lagrange

Math 5 Worksheet Solutions- Minmax and Lagrange. Find the local maximum and minimum values as well as the saddle point(s) of the function f(x, y) = e y (y x ). Solution. First we calculate the partial

### 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)

### Equations, Inequalities & Partial Fractions

Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

### Class Meeting # 1: Introduction to PDEs

MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x

### 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

### Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard

### Year 9 set 1 Mathematics notes, to accompany the 9H book.

Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

### Additional Topics in Linear Algebra Supplementary Material for Math 540. Joseph H. Silverman

Additional Topics in Linear Algebra Supplementary Material for Math 540 Joseph H Silverman E-mail address: jhs@mathbrownedu Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA Contents

### Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

### Determinants. Dr. Doreen De Leon Math 152, Fall 2015

Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.

### 1 Eigenvalues and Eigenvectors

Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

### 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

### 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

### Polynomial Invariants

Polynomial Invariants Dylan Wilson October 9, 2014 (1) Today we will be interested in the following Question 1.1. What are all the possible polynomials in two variables f(x, y) such that f(x, y) = f(y,

### A Brief Review of Elementary Ordinary Differential Equations

1 A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on

### Equations, Inequalities, Solving. and Problem AN APPLICATION

Equations, Inequalities, and Problem Solving. Solving Equations. Using the Principles Together AN APPLICATION To cater a party, Curtis Barbeque charges a \$0 setup fee plus \$ per person. The cost of Hotel