`Well, then,' the Gryphon went on, `if you don't know what to uglify is, you are a simpleton.'

Size: px
Start display at page:

Download "`Well, then,' the Gryphon went on, `if you don't know what to uglify is, you are a simpleton.'"

Transcription

1 La Back to the basics: Terms and techniques Rev 1/11/016 Reeling and Writhing, of course, to begin with, the Mock Turtle replied; and then the different branches of arithmetic -- Ambition, Distraction, Uglification, and Derision. `I never heard of "Uglification,"' Alice ventured to say. `What is it?' The Gryphon lifted up both its paws in surprise. `What! Never heard of uglifying!' it exclaimed. `You know what to beautify is, I suppose?' `Yes,' said Alice doubtfully: `it means--to--make--anything--prettier.' `Well, then,' the Gryphon went on, `if you don't know what to uglify is, you are a simpleton.' This lab is a veritable cornucopia of terminology and some basic ideas regarding partial derivatives and PDEs. You will find considerable overlap here with Multi-Variable Calculus. But the price of admission is: Review your notes for solutions to ODEs, especially the LINEAR and the EXACT ODEs. A scalar valued function has a vector derivative f f Let f ( xy, ) axyfor nonzero constant a. Then ay and ax. Thus the derivative of x y f(x, y), a scalar-valued function, is a vector that can be represented in the form:, f ( xy, ) fx( xy, ), fy( xy, ). In the example, that s just a<y, x>. x y Yes, you are in Wonderland! Scalar-valued functions have vector derivatives. But wait, there s more: If f(x,y) was a vector-valued function, say f(x, y) = <ax, by>, then we would need a matrix to represent all four first derivatives and there s a name for it:

2 a 0 The Jacobian Matrix J 0 b, (named for Carl Gustav Jacob Jacobi famous for having Jacob in his name twice; but more about him below). Can you determine how we obtained the elements in this Jacobian? We can take higher order and mixed partials of scalar functions: Let f(x,y) = axy: f f f f 0and 0 a x y x y y x And hence, a matrix for the second derivatives: Another example: x x y 0 a f( x, y) a 0 yx y Let f ( xy, ) ye x f x f x Then fx ye f y e and x y f x f x f f 4 0 xx ye fxy e fyx f yy x xy y Which leads to a Theorem: If f has continuous second order derivatives, f xy = f yx The order of differentiation may be reversed without changing the result. More Examples: See Implicit function definitions Sometimes it is easier to define a function of two or more variables implicitly (without a strict statement of u(x, y) = ax + by + cx + dy + e. It is of course possible to take regular derivatives implicitly (you knew that already); to be fair, whatever is good for regular derivatives must also be good for partials.

3 Example: Suppose we want the partial derivatives of u, where x y u a for a constant a. Note that we could write this explicitly as two functions: u a x y a x y,where. But why not just take the partial derivatives implicitly and solve: u :xu u 0or u x. x x x u After all, in the partial derivative with respect to x, y is constant. Do the same to find u y. Partial Practice Find the partial derivatives (you should do these by hand). 1. u = ln(x + y), find u u u,, x y xy. Find sin xy x cos( y) 3. If u = x + y and x = r cos, y = r sin (which you should recognize as polar u u coordinates), find, r Product rule with partial derivatives - what we re multiplying are functions! Suppose z = u v, where u = x y and v = cos (x y) u v Then: y ysin xy x x So that z u v x x x v u ycos xy xy sin xy z Find y. 3

4 The Chain Rule for Partial Derivatives dz f dx f dy If z = f (x,y) and x = g(t), y = h(t), then. Notice how the differentials that dt x dt y dt would seem to cancel out don t - because dx isn t the same as x. Example: f ( xy, ) cos( x y), xty, t df f x f y dt x t y t sin( t t t) sin( t t) Note answer in terms of t only. Important definition: The Total Derivative df ( x, y) f f dy Two variable case:, which can be written in differential form (known as dx x y dx f f the exact differential): df dx dy x y Three variables: df ( x, y, t) f f dx f dy. And on and on dt t x dt y dt Partial answers to partial practice above: 1. u = ln(x + y) : u 4 x, u xy x x y xy ( x y ) sin xy. y cos( xy)sec( y) x cos y u 3. u = x + y rcos rsin r More Practice Find the derivative of z. x y f( x, y, z) z yx with respect to x, with respect to y and with respect to 4

5 Some classy functions of more than one variable z = f(x,y) What are the partial derivatives,,,, x y xy x y f f f f f for each of the following? The monkey saddle z f x y x x y (, ) ( 4 ) So-named because there is a convenient place for the monkey s tail. Handkerchief 3 x z xy ( x y ) 3 Gently fluttering to the ground Want some curly fries with that? z = x sin x + y cos y, -< x,y < These functions are a good time to experiment with Plot3D; perhaps with some Manipulate[ ] as well For additional practice with functions of several variables (and some very useful plotting tricks), work through all examples in the Mathematica file Functions_Sev_Var.nb. Expand the cells marked with the downwards-pointing arrow on the right hand side by double clicking it. Then execute each command line and study the results. Complete the exercises in the sections marked Work on. 5

6 Here s some good news: Some PDEs are easy to solve, especially those that solve just like ODEs So that means: Review your notes on solutions to ODEs! Examples PDEs with only one partial derivative u 3x xy x Since u y doesn t show up, we can treat y as a constant and just integrate with respect to x. You should obtain u(x, y) = x 3 + x y / + F. What is F? Since it doesn t show up in the given equation for u x, F has to be something whose partial derivative with respect to x is 0 in other words, F is a function of y alone. It s like a constant of integration in an ODE; just constant with respect to x. Get used to that idea: constants that aren t always constant! Try some examples and verify that F(y) has no influence on the ability of our solution u(x, y) to fit the given PDE. nd order PDEs with only a single mixed derivative u 5x y xy u x xy We can just integrate twice: Integrating with respect to x, we obtain 5 F( y). Integrating with respect to y, we obtain x 3 x y uxy (, ) 5 y x Fy ( ) Gx ( ). Verify that this works! 3 Why didn t we have to integrate F(y)dy? It s an arbitrary function, to be determined by the initial and boundary conditions on the PDE. One arbitrary function is as good as any other. This PDE looks like an ODE, so maybe it is u xx u = 0 is just like the ODE u u = 0. However, when we integrate with respect to x, we again produce functions of y rather than constants of integration. Hence the solution u(x, y) = A(y) e x + B(y) e -x Verify that this works! 6

7 PDE with change of variables u xy = -u x Set p = u x, then p y = u xy = -p or p y /p = -1. This is now variables-separable in y: p = c(x) e -y and therefore u(x, y) = f(x) e -y + g(y) Verify that this works! Remember: constants of integration become functions of the other variable. Some Essential Terminology Analytic function that is just another way of saying that the function is differentiable with respect to the given variables. With PDEs, we just have lots more variables. Linear PDE all terms (both the function and/or its derivatives) are combined in a linear fashion, which may be stated quasi-formally as: If the PDE L(u, u x, u y ) = 0 and L is a linear operator, then the PDE is linear. L may contain a wide variety of functions and/or derivatives of the independent variables; just nothing like u x or u u x, but u xx is fine. Homogeneous polynomial all terms are in the same degree, ie x + xy + y = 0 Homogeneous PDE all terms including derivatives in the same degree (and usually the sum or product of functions and their derivatives = 0). The PDE L(u) = g(x) is homogeneous if g(x) = 0. It is always easier to solve a homogenous PDE than a non-homogenous PDE. Order the order of the PDE is the highest partial derivative in the equation. Higher order PDEs can still be linear. If you don t believe that, go back and reread what Linear PDE means. Coefficients anything multiplying a partial derivative or a function; they may be constants or variables or even functions. Boundary conditions (BCs) the solution function u and/or its derivatives have some known values on the edges of the domain of the problem. Also known as Boundary Values (BVs). 7

8 There are three general types of BV Problems (BVPs). 1. Dirichlet problems: Values of the solution function at a particular x are known for all times t: ie, u(0, t) = 0 or u(0, t) = f(t). For nd order PDEs like u xx + u yy = 0, the BC might be u(x,y) = a given function of x and y on the boundary of the region {x,y} where the problem is defined. Named for Johann Peter Gustav Lejeune Dirichlet, (famous for having 5 names) who was a student of Georg Ohm (of Ohm s Law fame), had a lifelong friend named Carl Jacobi (the Jacobian matrix guy with Jacob as his middle name). Dirichlet had a student named Georg Riemann (you ve heard of his sums, his hypothesis, his zeta function, etc). FYI: Gauss had this to say of Dirichlet: The total number of Dirichlet's publications is not large: jewels are not weighed on a grocery scale. See Neumann problems: Values of the partial derivatives of the solution function are known for a particular x and all times t: u x (0, t) = g 1 (t). Named for Carl Gottfried Neumann; he was a student of Otto Hesse, who in turn studied under Jacobi Robin problems: A mixture of the two other types in the form of a linear combination: u x (0, t) a u(0, t) = g (t). Named for Victor Gustave Robin, although we have uncovered all of his works (they are relatively few). Nowhere have we found him using the Robin boundary condition. Robin wrote a nice thesis in potential theory and also worked in thermodynamics. We have concluded that it is neither inappropriate nor especially appropriate that the third boundary condition now bears his name. 8

9 Note: BVPs that separately combine Dirichlet and Neumann conditions are sometimes known as Cauchy problems. Initial conditions (ICs) a specific type of Boundary Condition, in which the solution function u(x, t) and/or its derivatives have known values at time t = 0. ICs are also known as initial values (IVs); such problems are called initial value problems (IVPs). Consideration of various BCs and ICs are essential to the solution of PDEs that model real-world situations. Solutions often change dramatically with simple change of BCs. 9

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

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

More information

LS.6 Solution Matrices

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

More information

Class Meeting # 1: Introduction to PDEs

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

More information

Second Order Linear Partial Differential Equations. Part I

Second Order Linear Partial Differential Equations. Part I Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables; - point boundary value problems; Eigenvalues and Eigenfunctions Introduction

More information

CHAPTER 2. Eigenvalue Problems (EVP s) for ODE s

CHAPTER 2. Eigenvalue Problems (EVP s) for ODE s A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL EQUATIONS

More information

2 Integrating Both Sides

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

More information

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. 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

More information

1. First-order Ordinary Differential Equations

1. First-order Ordinary Differential Equations Advanced Engineering Mathematics 1. First-order ODEs 1 1. First-order Ordinary Differential Equations 1.1 Basic concept and ideas 1.2 Geometrical meaning of direction fields 1.3 Separable differential

More information

The Method of Partial Fractions Math 121 Calculus II Spring 2015

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

More information

DIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents

DIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition

More information

A Brief Review of Elementary Ordinary Differential Equations

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

More information

College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions

College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use

More information

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

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

More information

MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

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

More information

Microeconomic Theory: Basic Math Concepts

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

More information

Polynomial Invariants

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,

More information

Implicit Differentiation

Implicit Differentiation Implicit Differentiation Sometimes functions are given not in the form y = f(x) but in a more complicated form in which it is difficult or impossible to express y explicitly in terms of x. Such functions

More information

2.2 Separable Equations

2.2 Separable Equations 2.2 Separable Equations 73 2.2 Separable Equations An equation y = f(x, y) is called separable provided algebraic operations, usually multiplication, division and factorization, allow it to be written

More information

0.8 Rational Expressions and Equations

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

More information

Math 432 HW 2.5 Solutions

Math 432 HW 2.5 Solutions Math 432 HW 2.5 Solutions Assigned: 1-10, 12, 13, and 14. Selected for Grading: 1 (for five points), 6 (also for five), 9, 12 Solutions: 1. (2y 3 + 2y 2 ) dx + (3y 2 x + 2xy) dy = 0. M/ y = 6y 2 + 4y N/

More information

r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + 1 = 1 1 θ(t) 1 9.4.4 Write the given system in matrix form x = Ax + f ( ) sin(t) x y 1 0 5 z = dy cos(t)

r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + 1 = 1 1 θ(t) 1 9.4.4 Write the given system in matrix form x = Ax + f ( ) sin(t) x y 1 0 5 z = dy cos(t) Solutions HW 9.4.2 Write the given system in matrix form x = Ax + f r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + We write this as ( ) r (t) θ (t) = ( ) ( ) 2 r(t) θ(t) + ( ) sin(t) 9.4.4 Write the given system

More information

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

Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors 1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

More information

Partial Fractions. (x 1)(x 2 + 1)

Partial Fractions. (x 1)(x 2 + 1) Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

More information

10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS. The Jacobi Method

10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS. The Jacobi Method 578 CHAPTER 1 NUMERICAL METHODS 1. ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS As a numerical technique, Gaussian elimination is rather unusual because it is direct. That is, a solution is obtained after

More information

Homogeneous systems of algebraic equations. A homogeneous (ho-mo-geen -ius) system of linear algebraic equations is one in which

Homogeneous systems of algebraic equations. A homogeneous (ho-mo-geen -ius) system of linear algebraic equations is one in which Homogeneous systems of algebraic equations A homogeneous (ho-mo-geen -ius) 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

More information

Integrals of Rational Functions

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

More information

Limits and Continuity

Limits and Continuity Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function

More information

Separable First Order Differential Equations

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

More information

An Introduction to Partial Differential Equations

An Introduction to Partial Differential Equations An Introduction to Partial Differential Equations Andrew J. Bernoff LECTURE 2 Cooling of a Hot Bar: The Diffusion Equation 2.1. Outline of Lecture An Introduction to Heat Flow Derivation of the Diffusion

More information

5 Homogeneous systems

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

More information

Math 4310 Handout - Quotient Vector Spaces

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

More information

Homework #2 Solutions

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

More information

Solutions to Homework 10

Solutions to Homework 10 Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

More information

Differentiation and Integration

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

More information

DERIVATIVES AS MATRICES; CHAIN RULE

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

More information

1 Lecture: Integration of rational functions by decomposition

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.

More information

Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points

Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a

More information

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

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

More information

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1 Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

More information

Multi-variable Calculus and Optimization

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

More information

2.2 Derivative as a Function

2.2 Derivative as a Function 2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x

More information

Bindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8

Bindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8 Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e

More information

Constrained optimization.

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

More information

Solutions to Practice Problems for Test 4

Solutions to Practice Problems for Test 4 olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,

More information

Introduction to the Finite Element Method

Introduction to the Finite Element Method Introduction to the Finite Element Method 09.06.2009 Outline Motivation Partial Differential Equations (PDEs) Finite Difference Method (FDM) Finite Element Method (FEM) References Motivation Figure: cross

More information

A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors

More information

Linear and quadratic Taylor polynomials for functions of several variables.

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

More information

3. INNER PRODUCT SPACES

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.

More information

PYTHAGOREAN TRIPLES KEITH CONRAD

PYTHAGOREAN TRIPLES KEITH CONRAD PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

More information

3 Contour integrals and Cauchy s Theorem

3 Contour integrals and Cauchy s Theorem 3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of

More information

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

More information

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 3 EQUATIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.

More information

ORDINARY DIFFERENTIAL EQUATIONS

ORDINARY DIFFERENTIAL EQUATIONS ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.

More information

by the matrix A results in a vector which is a reflection of the given

by the matrix A results in a vector which is a reflection of the given Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

More information

Partial Fractions. p(x) q(x)

Partial Fractions. p(x) q(x) Partial Fractions Introduction to Partial Fractions Given a rational function of the form p(x) q(x) where the degree of p(x) is less than the degree of q(x), the method of partial fractions seeks to break

More information

x(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3

x(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3 CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract -

More information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

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

More information

1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes

1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.

More information

19.6. Finding a Particular Integral. Introduction. Prerequisites. Learning Outcomes. Learning Style

19.6. Finding a Particular Integral. Introduction. Prerequisites. Learning Outcomes. Learning Style Finding a Particular Integral 19.6 Introduction We stated in Block 19.5 that the general solution of an inhomogeneous equation is the sum of the complementary function and a particular integral. We have

More information

2 Session Two - Complex Numbers and Vectors

2 Session Two - Complex Numbers and Vectors PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar

More information

Unified Lecture # 4 Vectors

Unified Lecture # 4 Vectors Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

More information

Calculus 1: Sample Questions, Final Exam, Solutions

Calculus 1: Sample Questions, Final Exam, Solutions Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer.

More information

Metric Spaces. Chapter 7. 7.1. Metrics

Metric Spaces. Chapter 7. 7.1. Metrics Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

More information

F Matrix Calculus F 1

F Matrix Calculus F 1 F Matrix Calculus F 1 Appendix F: MATRIX CALCULUS TABLE OF CONTENTS Page F1 Introduction F 3 F2 The Derivatives of Vector Functions F 3 F21 Derivative of Vector with Respect to Vector F 3 F22 Derivative

More information

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t. LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing

More information

Differentiation of vectors

Differentiation of vectors Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where

More information

1 2 3 1 1 2 x = + x 2 + x 4 1 0 1

1 2 3 1 1 2 x = + x 2 + x 4 1 0 1 (d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which

More information

Inner Product Spaces

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

More information

7.6 Approximation Errors and Simpson's Rule

7.6 Approximation Errors and Simpson's Rule WileyPLUS: Home Help Contact us Logout Hughes-Hallett, Calculus: Single and Multivariable, 4/e Calculus I, II, and Vector Calculus Reading content Integration 7.1. Integration by Substitution 7.2. Integration

More information

tegrals as General & Particular Solutions

tegrals as General & Particular Solutions tegrals as General & Particular Solutions dy dx = f(x) General Solution: y(x) = f(x) dx + C Particular Solution: dy dx = f(x), y(x 0) = y 0 Examples: 1) dy dx = (x 2)2 ;y(2) = 1; 2) dy ;y(0) = 0; 3) dx

More information

The two dimensional heat equation

The two dimensional heat equation The two dimensional heat equation Ryan C. Trinity University Partial Differential Equations March 6, 2012 Physical motivation Consider a thin rectangular plate made of some thermally conductive material.

More information

SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA

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

More information

Vector and Matrix Norms

Vector and Matrix Norms Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

More information

Quotient Rings and Field Extensions

Quotient Rings and Field Extensions Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

More information

Math 265 (Butler) Practice Midterm II B (Solutions)

Math 265 (Butler) Practice Midterm II B (Solutions) Math 265 (Butler) Practice Midterm II B (Solutions) 1. Find (x 0, y 0 ) so that the plane tangent to the surface z f(x, y) x 2 + 3xy y 2 at ( x 0, y 0, f(x 0, y 0 ) ) is parallel to the plane 16x 2y 2z

More information

3. KINEMATICS IN TWO DIMENSIONS; VECTORS.

3. KINEMATICS IN TWO DIMENSIONS; VECTORS. 3. KINEMATICS IN TWO DIMENSIONS; VECTORS. Key words: Motion in Two Dimensions, Scalars, Vectors, Addition of Vectors by Graphical Methods, Tail to Tip Method, Parallelogram Method, Negative Vector, Vector

More information

Factoring Trinomials of the Form x 2 bx c

Factoring Trinomials of the Form x 2 bx c 4.2 Factoring Trinomials of the Form x 2 bx c 4.2 OBJECTIVES 1. Factor a trinomial of the form x 2 bx c 2. Factor a trinomial containing a common factor NOTE The process used to factor here is frequently

More information

SIXTY STUDY QUESTIONS TO THE COURSE NUMERISK BEHANDLING AV DIFFERENTIALEKVATIONER I

SIXTY STUDY QUESTIONS TO THE COURSE NUMERISK BEHANDLING AV DIFFERENTIALEKVATIONER I Lennart Edsberg, Nada, KTH Autumn 2008 SIXTY STUDY QUESTIONS TO THE COURSE NUMERISK BEHANDLING AV DIFFERENTIALEKVATIONER I Parameter values and functions occurring in the questions belowwill be exchanged

More information

CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

More information

Computational Geometry Lab: FEM BASIS FUNCTIONS FOR A TETRAHEDRON

Computational Geometry Lab: FEM BASIS FUNCTIONS FOR A TETRAHEDRON Computational Geometry Lab: FEM BASIS FUNCTIONS FOR A TETRAHEDRON John Burkardt Information Technology Department Virginia Tech http://people.sc.fsu.edu/ jburkardt/presentations/cg lab fem basis tetrahedron.pdf

More information

Chapter 20. Vector Spaces and Bases

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

More information

3.2 The Factor Theorem and The Remainder Theorem

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

More information

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

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.

More information

SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS

SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS A second-order linear differential equation has the form 1 Px d y dx dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. Equations of this type arise

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

General Framework for an Iterative Solution of Ax b. Jacobi s Method

General Framework for an Iterative Solution of Ax b. Jacobi s Method 2.6 Iterative Solutions of Linear Systems 143 2.6 Iterative Solutions of Linear Systems Consistent linear systems in real life are solved in one of two ways: by direct calculation (using a matrix factorization,

More information

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

More information

Numerical Methods for Differential Equations

Numerical Methods for Differential Equations Numerical Methods for Differential Equations Chapter 1: Initial value problems in ODEs Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical

More information

5 Scalings with differential equations

5 Scalings with differential equations 5 Scalings with differential equations 5.1 Stretched coordinates Consider the first-order linear differential equation df dx + f = 0. Since it is first order, we expect a single solution to the homogeneous

More information

Exam 1 Sample Question SOLUTIONS. y = 2x

Exam 1 Sample Question SOLUTIONS. y = 2x Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

More information

Nonhomogeneous Linear Equations

Nonhomogeneous Linear Equations Nonhomogeneous Linear Equations In this section we learn how to solve second-order nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form ay by cy G x where

More information

5.7 Maximum and Minimum Values

5.7 Maximum and Minimum Values 5.7 Maximum and Minimum Values Objectives Icandefinecriticalpoints. I know the di erence between local and absolute minimums/maximums. I can find local maximum(s), minimum(s), and saddle points for a given

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

Homework # 3 Solutions

Homework # 3 Solutions Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8

More information

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:

More information

Numerical Methods for Differential Equations

Numerical Methods for Differential Equations Numerical Methods for Differential Equations Course objectives and preliminaries Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical Analysis

More information

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

More information

An Introduction to Partial Differential Equations in the Undergraduate Curriculum

An Introduction to Partial Differential Equations in the Undergraduate Curriculum An Introduction to Partial Differential Equations in the Undergraduate Curriculum J. Tolosa & M. Vajiac LECTURE 11 Laplace s Equation in a Disk 11.1. Outline of Lecture The Laplacian in Polar Coordinates

More information

Higher Order Equations

Higher Order Equations Higher Order Equations We briefly consider how what we have done with order two equations generalizes to higher order linear equations. Fortunately, the generalization is very straightforward: 1. Theory.

More information