Scalar Valued Functions of Several Variables; the Gradient Vector


 Christal Wilkinson
 1 years ago
 Views:
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. Most functions of several variables commonly encountered have continuous partial 1
2 derivatives if we stay away from singularities. Thus the function f(x, y) = cos(x + y) sin(x y) is well defined and continuous except along the lines x y = k π, k an integer. The partial derivative functions are f sin(x + y) (x, y) = sin(x y) f sin(x + y) (x, y) = y sin(x y) + cos(x + y) sin 2 (x y) cos(x + y) sin 2 (x y) cos(x y), cos(x y). Each of these are also continuous functions of x, y except along the lines x y = k π, k an integer. Example Where the Partial Derivatives are Defined but not Continuous We consider the function defined by f(x, y) = x 3 + y 3 x 2 + y 2, (x, y) (0, 0), f(0, 0) = 0. Since this is defined in a nonstandard way at (0, 0), we should check on continuity at that point. If (x, y) (0, 0) lies in the square x b, y b, b > 0, we have x 3 + y 3 x 2 + y 2 x 3 + y 3 b (x 2 + y 2 ) = b. x 2 + y 2 x 2 + y 2 Thus as b 0, forcing (x, y) (0, 0), the value of f(x, y) tends to f(0, 0) = 0 and we conclude f(x, y) is continuous at (0, 0). On the xaxis f(x, 0) x so f (x, 0) 1; similarly on the yaxis. So the partial derivatives are defined on the axes; in particular they are defined and both equal to 1 at (0, 0). But for (x, y) (0, 0) we can compute ( x 3 + y 3 ) = 3x 2 x 2 + y 2 x 2 + y x 3 + y 3 2 (x 2 + y 2 ) 2 2x. Along a line y = a x this gives f (x, a x) = a 2(1 + a 3 ) 2 (1 + a 2 ). 2
3 Taking a = 1 we obtain f (x, x) 3/2 4/4 = 1/2 so, as we approach (0, 0) along this line f has a different value than if we approach the origin along either of the coordinate axes. We conclude f argument shows that f y is not continuous at the origin. A similar is also not continuous at the origin. The Gradient If we combine the partial derivatives (X 0 ), k = 1, 2,..., n into an ndimensional vector, in the obvious order, we obtain the gradient (vector) of the function φ(x) at 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 In R 3 we can write = φ(x) = φ(x 1, x 2,..., x n ) = (x, y, z)i + y φ(x) = ( (x, y, z), y ( (X), (X),..., ) (X). 1 2 n ) (x, y, z), (x, y, z) z (x, y, z)j + (x, y, z)k, i = (100), j = (010), k = (001). 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) = (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 3
4 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 ndimensional 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. k=1 x n If W is an ndimensional row vector then W X n k=1 w k x k. Gradients are usually written as row vectors; thus the preceding implies that φ(x 0 )X n k=1 (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 ) = φ(x 0 ) + n k=1 (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). 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 ). 4
5 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 2 φ(x, y, z) = (2x 3y 2 4z 3 ); φ(1, 1, 1) = 3. 4 Accordingly, the linear approximation to φ(x, y, z) as given by the above formula is given by the function x 1 w = 3 + (2 3 4 ) y 1 = 3 + 2(x 1) + 3(y 1) + 4(z 1) z 1 = 6 + 2x + 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(1.1) + 3(.9) + 4(1.05) = 3.1, whereas the actual value is φ(1.1,.9, 1.05) = (1.1) 2 + (.9) 3 + (1.05) 4 = 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 ). 5
6 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) + (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 0, y 0 )(x x 0 ) ( ) ( (x, η) y y (x 0, y 0 ) (y y 0 ) + (ξ, y 0) ) (x 0, y 0 ) (x x 0 ) where γ(x 0, X) is the row vector = φ(x 0, y 0 )(X X 0 ) + γ(x 0, X)(X X 0 ), γ(x 0, X) = ( y (x, η) y (x 0, y 0 ) (ξ, y 0) (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 γ(x 0, X) = X X 0 lim ( X X y 0 (x, η) y (x 0, y 0 ) (ξ, y 0) (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 = 0, a relationship which we express by saying that 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 6
7 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. We conclude with the following version of the Chain Rule. Proposition 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(t))x (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 )(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 ))(X (t 0 )(t t 0 ) + o( t t 0 ). Dividing by t t 0 we then have 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 d dt φ(x(t)) = φ(x(t 0 ))X (t 0 ). t=t0 Since t 0 could be any value of t, the result follows. 7
8 Application: The Directional Derivative Let φ(x) be a continuously differentiable scalar valued function of the nvector variable X. Let X 0 be a point in its domain and let U be a unit vector. Then we can construct the line through X 0 in the direction of U via X(t) = X 0 + t U, t 0. Clearly X(0) = X 0. We form the composite function φ(x(t)) and differentiate with respect to t using the chain rule: d dt φ(x 0 + t U) = φ(x 0 + t U) d dt (X 0 + t U) = φ(x 0 + t U)U. This is the rate of change of φ along the line described by X(t). Taking t = 0, the rate of change of φ in the direction of this line, i.e., in the direction of U, at X 0 = X(0), is seen to be φ(x 0 ) U. This is called the directional derivative of φ in the direction of U at the point X 0. The notation φ U U denote this quantity. Example is often used to For the function φ(x, y, z) = x 2 + 4y 2 + 9z 2 the directional derivative in the direction given by U = ( 1/ 6, 2/ 6, 1/ 6) at the point X 0 = (1, 1, 1) is / 6 2/ 6 1/ 6 = 1 (2( 1) + 8(2) + 18( 1)) = Properties of the Directional Derivative the dot, or inner product we have From the corresponding property of φ(x 0 )U = φ(x 0 ) U cos θ = φ(x 0 ) cosθ, where θ is the angle between the gradient vector φ(x 0 ) and U. If the gradient vector φ(x 0 ) 0 it is clear that this quantity reaches its maximum, φ(x 0 ), when θ = 0, in which case U = U + φ(x 0), and reaches its minimum, φ(x φ(x 0 ) 0), when θ = ±π, in which case U = U φ(x 0) φ(x 0. The direction corresponding to ) U + is called the steepest ascent direction for φ at X 0 while the direction corresponding to U is called the steepest descent direction. 8
Scalar Valued Functions of Several Variables; the Gradient Vector
Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,
More informationSection 12.6: Directional Derivatives and the Gradient Vector
Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate
More informationMATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.
MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin
More informationRecall 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
More informationTaylor 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
More informationDERIVATIVES AS MATRICES; CHAIN RULE
DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Realvalued 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 informationLimits 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 informationPractice Problems for Midterm 2
Practice Problems for Midterm () For each of the following, find and sketch the domain, find the range (unless otherwise indicated), and evaluate the function at the given point P : (a) f(x, y) = + 4 y,
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationDifferentiation of vectors
Chapter 4 Differentiation of vectors 4.1 Vectorvalued 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 informationVectors, 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
More informationEngineering Math II Spring 2015 Solutions for Class Activity #2
Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the xaxis. Then find
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional 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 informationSolutions to Vector Calculus Practice Problems
olutions to Vector alculus Practice Problems 1. Let be the region in determined by the inequalities x + y 4 and y x. Evaluate the following integral. sinx + y ) da Answer: The region looks like y y x x
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More information2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair
2.1 J.A.Beachy 1 2.1 Functions from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 21. The Vertical Line Test from calculus says that a curve in the xyplane
More informationSection 5: The Jacobian matrix and applications. S1: Motivation S2: Jacobian matrix + differentiability S3: The chain rule S4: Inverse functions
Section 5: The Jacobian matri and applications. S1: Motivation S2: Jacobian matri + differentiabilit S3: The chain rule S4: Inverse functions Images from Thomas calculus b Thomas, Wier, Hass & Giordano,
More informationMath 497C Sep 9, Curves and Surfaces Fall 2004, PSU
Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 15 sometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics We say
More informationHOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba
HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain
More information1 Review of complex numbers
1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely
More informationClass 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 information3.8 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
More informationSome 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
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationby 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 yaxis We observe that
More informationChapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis
Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More informationSOLUTIONS. 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 informationMath 5311 Gateaux differentials and Frechet derivatives
Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.
More informationVector 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
More informationMath 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z).
Math 29 Solutions to Assignment 7. Find the gradient vector field of the following functions: a fx, y lnx + 2y; b fx, y, z x cosy/z. Solution. a f x x + 2y, f 2 y x + 2y. Thus, the gradient vector field
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrixvector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More informationF 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 informationcorrectchoice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More informationInverse Functions and Logarithms
Section 3. Inverse Functions and Logarithms 1 Kiryl Tsishchanka Inverse Functions and Logarithms DEFINITION: A function f is called a onetoone function if it never takes on the same value twice; that
More informationUnified 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 informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More information106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM 5.1.1 Fermat s Theorem f is differentiable at a, then f (a) = 0.
5 Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function
More informationContents. Introduction and Notes pages 23 (These are important and it s only 2 pages ~ please take the time to read them!)
Page Contents Introduction and Notes pages 23 (These are important and it s only 2 pages ~ please take the time to read them!) Systematic Search for a Change of Sign (Decimal Search) Method Explanation
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More informationUsing a table of derivatives
Using a table of derivatives In this unit we construct a Table of Derivatives of commonly occurring functions. This is done using the knowledge gained in previous units on differentiation from first principles.
More informationTOPIC 3: CONTINUITY OF FUNCTIONS
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let
More informationSec 4.1 Vector Spaces and Subspaces
Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common
More information3 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 information4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
More informationSolutions 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 informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationSection 2.7 OnetoOne Functions and Their Inverses
Section. OnetoOne Functions and Their Inverses OnetoOne Functions HORIZONTAL LINE TEST: A function is onetoone if and only if no horizontal line intersects its graph more than once. EXAMPLES: 1.
More informationA linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form
Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...
More informationSolutions 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
More informationx(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 informationWASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,
More informationVector 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 nonempty
More informationParametric Curves, Vectors and Calculus. Jeff Morgan Department of Mathematics University of Houston
Parametric Curves, Vectors and Calculus Jeff Morgan Department of Mathematics University of Houston jmorgan@math.uh.edu Online Masters of Arts in Mathematics at the University of Houston http://www.math.uh.edu/matweb/grad_mam.htm
More informationTRANSFORMATIONS OF RANDOM VARIABLES
TRANSFORMATIONS OF RANDOM VARIABLES 1. INTRODUCTION 1.1. Definition. We are often interested in the probability distributions or densities of functions of one or more random variables. Suppose we have
More informationv 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product)
0.1 Cross Product The dot product of two vectors is a scalar, a number in R. Next we will define the cross product of two vectors in 3space. This time the outcome will be a vector in 3space. Definition
More information1 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 informationChapter 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
More informationL 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:
More informationVector has a magnitude and a direction. Scalar has a magnitude
Vector has a magnitude and a direction Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude a brick on a table Vector has a magnitude and a direction Scalar has a magnitude
More informationLinearly Independent Sets and Linearly Dependent Sets
These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for inclass presentation
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationLimit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)
SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationMATH 1231 S2 2010: Calculus. Section 1: Functions of severable variables.
MATH 1231 S2 2010: Calculus For use in Dr Chris Tisdell s lectures Section 1: Functions of severable variables. Created and compiled by Chris Tisdell S1: Motivation S2: Function of two variables S3: Visualising
More informationDouble integrals. Notice: this material must not be used as a substitute for attending the lectures
ouble integrals Notice: this material must not be used as a substitute for attending the lectures . What is a double integral? Recall that a single integral is something of the form b a f(x) A double integral
More informationMATH 304 Linear Algebra Lecture 24: Scalar product.
MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent
More information3. Double Integrals 3A. Double Integrals in Rectangular Coordinates
3. Double Integrals 3A. Double Integrals in ectangular Coordinates 3A1 Evaluate each of the following iterated integrals: c) 2 1 1 1 x 2 (6x 2 +2y)dydx b) x 2x 2 ydydx d) π/2 π 1 u (usint+tcosu)dtdu u2
More informationMATH 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
More informationTaylor 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
More information9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is socalled because when the scalar product of
More informationMATH 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 informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationA 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 informationThis function is symmetric with respect to the yaxis, so I will let  /2 /2 and multiply the area by 2.
INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More informationIntroduction 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 information2.1: MATRIX OPERATIONS
.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More information15.1. Vector Analysis. Vector Fields. Objectives. Vector Fields. Vector Fields. Vector Fields. ! Understand the concept of a vector field.
15 Vector Analysis 15.1 Vector Fields Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives! Understand the concept of a vector field.! Determine
More informationMATRIX 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 +
More informationThe determinant of a skewsymmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14
4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with
More informationSolutions 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 informationLinear Algebra: Vectors
A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector
More informationItems related to expected use of graphing technology appear in bold italics.
 1  Items related to expected use of graphing technology appear in bold italics. Investigating the Graphs of Polynomial Functions determine, through investigation, using graphing calculators or graphing
More informationMATRIX 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
More informationVECTOR CALCULUS: USEFUL STUFF Revision of Basic Vectors
Prof. S.M. Tobias Jan 2009 VECTOR CALCULUS: USEFUL STUFF Revision of Basic Vectors A scalar is a physical quantity with magnitude only A vector is a physical quantity with magnitude and direction A unit
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More informationSimilarity 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
More informationLinear 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
More informationMatrix Differentiation
1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have
More informationx 2 + y 2 = 25 and try to solve for y in terms of x, we get 2 new equations y = 25 x 2 and y = 25 x 2.
Lecture : Implicit differentiation For more on the graphs of functions vs. the graphs of general equations see Graphs of Functions under Algebra/Precalculus Review on the class webpage. For more on graphing
More informationVector algebra Christian Miller CS Fall 2011
Vector algebra Christian Miller CS 354  Fall 2011 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority
More information1 Scalars, Vectors and Tensors
DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical
More informationUnderstanding 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
More informationWe call this set an ndimensional 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
More informationHomework # 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 informationDefinition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =
Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a
More information