Differentiation of Multivariable Functions

Total Page:16

File Type:pdf, Size:1020Kb

Differentiation of Multivariable Functions CHAPTER 3 Differentiation of Multivariable Functions 16. Functions of Several Variables The concept of a function of several variables can be qualitatively un- derstood from simple examples in everyday life. The temperature in a room may vary from point to point. A point in space can be defined by an or- dered triple of numbers that are coordinates of the point in some coordinate system, say, (x,y,z). Measurements of the temperature at every point from a set D in space assign a real number T (the temperature) to every point of D. The dependence of T on coordinates of the point is indicated by writing T = T (x,y,z). Similarly, the concentration of a chemical can depend on a point in space. In addition, if the chemical reacts with other chemicals, its concentration at a point may also change with time. In this case, the concen- tration C = C(x,y,z,t) depends on four variables, three spatial coordinates and the time t. In general, if the value of a quantity f depends on values of several other quantities, say, x1, x2,..., xm, this dependence is indicated by writing f = f(x1,x2,...,xm). In other words, f = f(x1,x2,...,xm) indicates a rule that assigns a unique real number f to each ordered m-tuple of real numbers (x1,x2,...,xm): f : (x ,x ,...,x ) f(x ,x ,...,x ) 1 2 m → 1 2 m Each number in the m-tuple may be of a different nature and measured in different units. In the above example, the concentration depends on ordered quadruples (x,y,z,t), where x, y, and z are the coordinates of a point in space (measured in units of length) and t is time (measured in units of time). To analyze properties of functions of several variables, a notion of a distance between two ordered m tuples is needed. For example, a rate of change of a function is naturally− defined as the difference of values of the function at two points divided by the distance between them. This allows us to determine that one function changes more rapidly than the other. In what follows, functions on Euclidean spaces will be studied. In other words, the distance between two ordered m tuples (or two arguments of a function) is assumed to be the Euclidean distance.− To simplify notations, the argument of a function of several variables will often be written in the vector form f(x ,x ,...,x ) = f(r) , r Rm . 1 2 m ∈ The value of a function at a particular point P of a Euclidean space will also be denoted by f(P ) to emphasize that this value is independent of the choice of a coordinate system in which coordinates of P are given. For 241 242 3. DIFFERENTIATION OF MULTIVARIABLE FUNCTIONS f D D f y P P f(P ) f(P ) x R f R f Figure 16.1. Left: A function f of two variables is a rule, f : P f(P ), that assigns a number f(P ) to every point P of a→ planar region D. The set R of all numbers f(P ) is the range of f. The region D is the domain of f. Right: A function f of three variables is a rule that assigns a number f(P ) to every point P of a solid region D. example, a temperature in a room T = T (P ) depends on a point P , but the coordinates of P (the ordered triples of numbers) are different in different coordinate systems. 16.1. Real-Valued Functions of Several Variables. Definition 16.1. (Real-Valued Function of Several Variables). Let D be a set of ordered m-tuples of real numbers (x1,x2,...,xm). A func- tion f of m variables is a rule that assigns to each m tuple in the set D − a unique real number denoted by f(x1,x2,...,xm). The set D is the do- main of f, and its range is the set of values that f takes on it, that is, f(x ,x ,...,x ) (x ,x ,...,x ) D . { 1 2 m | 1 2 m ∈ } This definition is illustrated in Fig. 16.1. The rule may be defined by different means. For example, if D is a collection of N points Pi, i = 1, 2,...,N, then a function f can be defined by a table (Pi, f(Pi)), where f(Pi) is the value of f at Pi. A function f can be defined geometrically. For example, the height of a mountain relative to sea level is a function of its position on the globe. So the height is a function of two variables, the longitude and latitude. A function can be defined by an algebraic rule that prescribes algebraic operations to be carried out with real numbers in any n-tuple to obtain the value of the function. For example, f(x,y,z) = x2 y + z3 . − The value of this function at (1, 2, 3) is f(1, 2, 3)= 12 2+33 = 26 . − Unless specified otherwise, the domain of a function defined by an algebraic rule is the set of m-tuples for which the rule makes sense. Example 16.1. Find the domain and the range of the function of two variables f(x, y) = ln(1 x2 y2). − − 16. FUNCTIONS OF SEVERAL VARIABLES 243 Solution: The logarithmis defined for any strictly positive number. There- fore, the doublets (x, y) must be such that 1 x2 y2 > 0 or x2 + y2 < 1. Hence, − − D = (x, y) x2 + y2 < 1 . { | } Since any doublet (x, y) can be uniquely associated with a point on a plane, the set D can be given a geometrical description as a disk of radius 1 whose boundary, the circle x2 + y2 = 1, is not included in D. For any point in the interior of the disk, the argument of the logarithm lies in the interval 0 < 1 x2 y2 1. So the range of f is the set of values of the logarithm in the− interval− (0≤, 1], which is < f 0. −∞ ≤ Example 16.2. Find the domain and the range of the function of three variables f(x,y,z) = x2 z x2 y2. − − Solution: The square rootp is defined only for nonnegative numbers. There- fore, ordered triples (x,y,z) must be such that z x2 y2 0, that is, − − ≥ D = (x,y,z) z x2 + y2 . { | ≥ } This set can be given a geometrical description as a point set in space because any triple can be associated with a unique point in space. The equation z = x2 + y2 describes a circular paraboloid. So the domain is the spatial (solid) region containing points that lie on or above the paraboloid. The function is non- negative. By fixing x and y and increasing z, one can see that the value of f can be any positive number. So the range is 0 f(x,y,z) < . The domain of a function of m variables is≤ viewed as a subset∞ of an m-dimensional Euclidean space. For example, the domain of the function f(r)=(1 x2 x2 x2 )1/2 = (1 r 2)1/2 , − 1 − 2 −···− m −k k where r = x1,x2,...,xm , is the set of points inthe m-dimensional Euclidean space whoseh distance fromi the origin (the zero vector) does not exceed 1, D = r Rm r 1 ; { ∈ | k k ≤ } that is, it is a closed m-dimensional ball of radius 1. So the domain of a multivariable function defined by an algebraic rule can be described by conditions on the components of the ordered m-tuple r under which the rule makes sense. 16.2. The Graph of a Function of Two Variables. The graph of a function of one variable f(x) is the set of points (x, y) of a plane such that y = f(x) , x D ∈ where the domain D is a collection of points on the x axis. The graph is obtained by moving a point of the domain parallel to the y axis by an amount determined by the value of the function y = f(x). The graph provides a 244 3. DIFFERENTIATION OF MULTIVARIABLE FUNCTIONS z (x,y,z) z z = f(x, y) 1 y D y 2 3 (x,y, 0) D x x Figure 16.2. Left: The graph of a function of two vari- ables is the surface defined by the equation z = f(x, y). It is obtained from the domain D of f by moving each point (x,y, 0) in D along the z axis to the point (x,y,f(x, y)). Right: The graph of the function studied in Example 16.3. useful picture of the behavior of the function. The idea can be extended to functions of two variables. Definition 16.2. (Graph of a Function of Two Variables). The graph of a function f(x, y) with domain D is a collection of points (x,y,z) in space such that z = f(x, y) , (x, y) D. ∈ The domain D is a set of points in the xy plane. The graph is then obtained by moving each point of D parallel to the z axis by an amount equal to the corresponding value of the function z = f(x, y). If D is a portion of the plane, then the graph of f is generally a surface (see Fig. 16.2 (left panel)). One can think of the graph as “mountains” of height f(x, y) on the xy plane.
Recommended publications
  • Calculus for the Life Sciences I Lecture Notes – Limits, Continuity, and the Derivative
    Limits Continuity Derivative Calculus for the Life Sciences I Lecture Notes – Limits, Continuity, and the Derivative Joseph M. Mahaffy, [email protected] Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State University San Diego, CA 92182-7720 http://www-rohan.sdsu.edu/∼jmahaffy Spring 2013 Lecture Notes – Limits, Continuity, and the Deriv Joseph M. Mahaffy, [email protected] — (1/24) Limits Continuity Derivative Outline 1 Limits Definition Examples of Limit 2 Continuity Examples of Continuity 3 Derivative Examples of a derivative Lecture Notes – Limits, Continuity, and the Deriv Joseph M. Mahaffy, [email protected] — (2/24) Limits Definition Continuity Examples of Limit Derivative Introduction Limits are central to Calculus Lecture Notes – Limits, Continuity, and the Deriv Joseph M. Mahaffy, [email protected] — (3/24) Limits Definition Continuity Examples of Limit Derivative Introduction Limits are central to Calculus Present definitions of limits, continuity, and derivative Lecture Notes – Limits, Continuity, and the Deriv Joseph M. Mahaffy, [email protected] — (3/24) Limits Definition Continuity Examples of Limit Derivative Introduction Limits are central to Calculus Present definitions of limits, continuity, and derivative Sketch the formal mathematics for these definitions Lecture Notes – Limits, Continuity, and the Deriv Joseph M. Mahaffy, [email protected] — (3/24) Limits Definition Continuity Examples of Limit Derivative Introduction Limits
    [Show full text]
  • Hegel on Calculus
    HISTORY OF PHILOSOPHY QUARTERLY Volume 34, Number 4, October 2017 HEGEL ON CALCULUS Ralph M. Kaufmann and Christopher Yeomans t is fair to say that Georg Wilhelm Friedrich Hegel’s philosophy of Imathematics and his interpretation of the calculus in particular have not been popular topics of conversation since the early part of the twenti- eth century. Changes in mathematics in the late nineteenth century, the new set-theoretical approach to understanding its foundations, and the rise of a sympathetic philosophical logic have all conspired to give prior philosophies of mathematics (including Hegel’s) the untimely appear- ance of naïveté. The common view was expressed by Bertrand Russell: The great [mathematicians] of the seventeenth and eighteenth cen- turies were so much impressed by the results of their new methods that they did not trouble to examine their foundations. Although their arguments were fallacious, a special Providence saw to it that their conclusions were more or less true. Hegel fastened upon the obscuri- ties in the foundations of mathematics, turned them into dialectical contradictions, and resolved them by nonsensical syntheses. .The resulting puzzles [of mathematics] were all cleared up during the nine- teenth century, not by heroic philosophical doctrines such as that of Kant or that of Hegel, but by patient attention to detail (1956, 368–69). Recently, however, interest in Hegel’s discussion of calculus has been awakened by an unlikely source: Gilles Deleuze. In particular, work by Simon Duffy and Henry Somers-Hall has demonstrated how close Deleuze and Hegel are in their treatment of the calculus as compared with most other philosophers of mathematics.
    [Show full text]
  • Lecture 9: Partial Derivatives
    Math S21a: Multivariable calculus Oliver Knill, Summer 2016 Lecture 9: Partial derivatives ∂ If f(x,y) is a function of two variables, then ∂x f(x,y) is defined as the derivative of the function g(x) = f(x,y) with respect to x, where y is considered a constant. It is called the partial derivative of f with respect to x. The partial derivative with respect to y is defined in the same way. ∂ We use the short hand notation fx(x,y) = ∂x f(x,y). For iterated derivatives, the notation is ∂ ∂ similar: for example fxy = ∂x ∂y f. The meaning of fx(x0,y0) is the slope of the graph sliced at (x0,y0) in the x direction. The second derivative fxx is a measure of concavity in that direction. The meaning of fxy is the rate of change of the slope if you change the slicing. The notation for partial derivatives ∂xf,∂yf was introduced by Carl Gustav Jacobi. Before, Josef Lagrange had used the term ”partial differences”. Partial derivatives fx and fy measure the rate of change of the function in the x or y directions. For functions of more variables, the partial derivatives are defined in a similar way. 4 2 2 4 3 2 2 2 1 For f(x,y)= x 6x y + y , we have fx(x,y)=4x 12xy ,fxx = 12x 12y ,fy(x,y)= − − − 12x2y +4y3,f = 12x2 +12y2 and see that f + f = 0. A function which satisfies this − yy − xx yy equation is also called harmonic. The equation fxx + fyy = 0 is an example of a partial differential equation: it is an equation for an unknown function f(x,y) which involves partial derivatives with respect to more than one variables.
    [Show full text]
  • Limits and Derivatives 2
    57425_02_ch02_p089-099.qk 11/21/08 10:34 AM Page 89 FPO thomasmayerarchive.com Limits and Derivatives 2 In A Preview of Calculus (page 3) we saw how the idea of a limit underlies the various branches of calculus. Thus it is appropriate to begin our study of calculus by investigating limits and their properties. The special type of limit that is used to find tangents and velocities gives rise to the central idea in differential calcu- lus, the derivative. We see how derivatives can be interpreted as rates of change in various situations and learn how the derivative of a function gives information about the original function. 89 57425_02_ch02_p089-099.qk 11/21/08 10:35 AM Page 90 90 CHAPTER 2 LIMITS AND DERIVATIVES 2.1 The Tangent and Velocity Problems In this section we see how limits arise when we attempt to find the tangent to a curve or the velocity of an object. The Tangent Problem The word tangent is derived from the Latin word tangens, which means “touching.” Thus t a tangent to a curve is a line that touches the curve. In other words, a tangent line should have the same direction as the curve at the point of contact. How can this idea be made precise? For a circle we could simply follow Euclid and say that a tangent is a line that intersects the circle once and only once, as in Figure 1(a). For more complicated curves this defini- tion is inadequate. Figure l(b) shows two lines and tl passing through a point P on a curve (a) C.
    [Show full text]
  • Policy Gradient
    Lecture 7: Policy Gradient Lecture 7: Policy Gradient David Silver Lecture 7: Policy Gradient Outline 1 Introduction 2 Finite Difference Policy Gradient 3 Monte-Carlo Policy Gradient 4 Actor-Critic Policy Gradient Lecture 7: Policy Gradient Introduction Policy-Based Reinforcement Learning In the last lecture we approximated the value or action-value function using parameters θ, V (s) V π(s) θ ≈ Q (s; a) Qπ(s; a) θ ≈ A policy was generated directly from the value function e.g. using -greedy In this lecture we will directly parametrise the policy πθ(s; a) = P [a s; θ] j We will focus again on model-free reinforcement learning Lecture 7: Policy Gradient Introduction Value-Based and Policy-Based RL Value Based Learnt Value Function Implicit policy Value Function Policy (e.g. -greedy) Policy Based Value-Based Actor Policy-Based No Value Function Critic Learnt Policy Actor-Critic Learnt Value Function Learnt Policy Lecture 7: Policy Gradient Introduction Advantages of Policy-Based RL Advantages: Better convergence properties Effective in high-dimensional or continuous action spaces Can learn stochastic policies Disadvantages: Typically converge to a local rather than global optimum Evaluating a policy is typically inefficient and high variance Lecture 7: Policy Gradient Introduction Rock-Paper-Scissors Example Example: Rock-Paper-Scissors Two-player game of rock-paper-scissors Scissors beats paper Rock beats scissors Paper beats rock Consider policies for iterated rock-paper-scissors A deterministic policy is easily exploited A uniform random policy
    [Show full text]
  • Introduction to Shape Optimization
    Introduction to Shape optimization Noureddine Igbida1 1Institut de recherche XLIM, UMR-CNRS 6172, Facult´edes Sciences et Techniques, Universit´ede Limoges 123, Avenue Albert Thomas 87060 Limoges, France. Email : [email protected] Preliminaries on PDE 1 Contents 1. PDE ........................................ 2 2. Some differentiation and differential operators . 3 2.1 Gradient . 3 2.2 Divergence . 4 2.3 Curl . 5 2 2.4 Remarks . 6 2.5 Laplacian . 7 2.6 Coordinate expressions of the Laplacian . 10 3. Boundary value problem . 12 4. Notion of solution . 16 1. PDE A mathematical model is a description of a system using mathematical language. The process of developing a mathematical model is termed mathematical modelling (also written model- ing). Mathematical models are used in many area, as in the natural sciences (such as physics, biology, earth science, meteorology), engineering disciplines (e.g. computer science, artificial in- telligence), in the social sciences (such as economics, psychology, sociology and political science); Introduction to Shape optimization N. Igbida physicists, engineers, statisticians, operations research analysts and economists. Among this mathematical language we have PDE. These are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic. In this section, we give some basic example of elliptic partial differential equation (PDE) of second order : standard Laplacian and Laplacian with variable coefficients.
    [Show full text]
  • Vector Calculus and Differential Forms with Applications To
    Vector Calculus and Differential Forms with Applications to Electromagnetism Sean Roberson May 7, 2015 PREFACE This paper is written as a final project for a course in vector analysis, taught at Texas A&M University - San Antonio in the spring of 2015 as an independent study course. Students in mathematics, physics, engineering, and the sciences usually go through a sequence of three calculus courses before go- ing on to differential equations, real analysis, and linear algebra. In the third course, traditionally reserved for multivariable calculus, stu- dents usually learn how to differentiate functions of several variable and integrate over general domains in space. Very rarely, as was my case, will professors have time to cover the important integral theo- rems using vector functions: Green’s Theorem, Stokes’ Theorem, etc. In some universities, such as UCSD and Cornell, honors students are able to take an accelerated calculus sequence using the text Vector Cal- culus, Linear Algebra, and Differential Forms by John Hamal Hubbard and Barbara Burke Hubbard. Here, students learn multivariable cal- culus using linear algebra and real analysis, and then they generalize familiar integral theorems using the language of differential forms. This paper was written over the course of one semester, where the majority of the book was covered. Some details, such as orientation of manifolds, topology, and the foundation of the integral were skipped to save length. The paper should still be readable by a student with at least three semesters of calculus, one course in linear algebra, and one course in real analysis - all at the undergraduate level.
    [Show full text]
  • Limits of Functions
    Chapter 2 Limits of Functions In this chapter, we define limits of functions and describe some of their properties. 2.1. Limits We begin with the ϵ-δ definition of the limit of a function. Definition 2.1. Let f : A ! R, where A ⊂ R, and suppose that c 2 R is an accumulation point of A. Then lim f(x) = L x!c if for every ϵ > 0 there exists a δ > 0 such that 0 < jx − cj < δ and x 2 A implies that jf(x) − Lj < ϵ. We also denote limits by the `arrow' notation f(x) ! L as x ! c, and often leave it to be implicitly understood that x 2 A is restricted to the domain of f. Note that we exclude x = c, so the function need not be defined at c for the limit as x ! c to exist. Also note that it follows directly from the definition that lim f(x) = L if and only if lim jf(x) − Lj = 0: x!c x!c Example 2.2. Let A = [0; 1) n f9g and define f : A ! R by x − 9 f(x) = p : x − 3 We claim that lim f(x) = 6: x!9 To prove this, let ϵ >p 0 be given. For x 2 A, we have from the difference of two squares that f(x) = x + 3, and p x − 9 1 jf(x) − 6j = x − 3 = p ≤ jx − 9j: x + 3 3 Thus, if δ = 3ϵ, then jx − 9j < δ and x 2 A implies that jf(x) − 6j < ϵ.
    [Show full text]
  • MATH M25BH: Honors: Calculus with Analytic Geometry II 1
    MATH M25BH: Honors: Calculus with Analytic Geometry II 1 MATH M25BH: HONORS: CALCULUS WITH ANALYTIC GEOMETRY II Originator brendan_purdy Co-Contributor(s) Name(s) Abramoff, Phillip (pabramoff) Butler, Renee (dbutler) Balas, Kevin (kbalas) Enriquez, Marcos (menriquez) College Moorpark College Attach Support Documentation (as needed) Honors M25BH.pdf MATH M25BH_state approval letter_CCC000621759.pdf Discipline (CB01A) MATH - Mathematics Course Number (CB01B) M25BH Course Title (CB02) Honors: Calculus with Analytic Geometry II Banner/Short Title Honors: Calc/Analy Geometry II Credit Type Credit Start Term Fall 2021 Catalog Course Description Reviews integration. Covers area, volume, arc length, surface area, centers of mass, physics applications, techniques of integration, improper integrals, sequences, series, Taylor’s Theorem, parametric equations, polar coordinates, and conic sections with translations. Honors work challenges students to be more analytical and creative through expanded assignments and enrichment opportunities. Additional Catalog Notes Course Credit Limitations: 1. Credit will not be awarded for both the honors and regular versions of a course. Credit will be awarded only for the first course completed with a grade of "C" or better or "P". Honors Program requires a letter grade. 2. MATH M16B and MATH M25B or MATH M25BH combined: maximum one course for transfer credit. Taxonomy of Programs (TOP) Code (CB03) 1701.00 - Mathematics, General Course Credit Status (CB04) D (Credit - Degree Applicable) Course Transfer Status (CB05)
    [Show full text]
  • Using Surface Integrals for Checking the Archimedes' Law of Buoyancy
    Using surface integrals for checking the Archimedes’ law of buoyancy F M S Lima Institute of Physics, University of Brasilia, P.O. Box 04455, 70919-970, Brasilia-DF, Brazil E-mail: [email protected] Abstract. A mathematical derivation of the force exerted by an inhomogeneous (i.e., compressible) fluid on the surface of an arbitrarily-shaped body immersed in it is not found in literature, which may be attributed to our trust on Archimedes’ law of buoyancy. However, this law, also known as Archimedes’ principle (AP), does not yield the force observed when the body is in contact to the container walls, as is more evident in the case of a block immersed in a liquid and in contact to the bottom, in which a downward force that increases with depth is observed. In this work, by taking into account the surface integral of the pressure force exerted by a fluid over the surface of a body, the general validity of AP is checked. For a body fully surrounded by a fluid, homogeneous or not, a gradient version of the divergence theorem applies, yielding a volume integral that simplifies to an upward force which agrees to the force predicted by AP, as long as the fluid density is a continuous function of depth. For the bottom case, this approach yields a downward force that increases with depth, which contrasts to AP but is in agreement to experiments. It also yields a formula for this force which shows that it increases with the area of contact. PACS numbers: 01.30.mp, 01.55.+b, 47.85.Dh Submitted to: Eur.
    [Show full text]
  • Mathematics for Economics Anthony Tay 7. Limits of a Function The
    Mathematics for Economics Anthony Tay 7. Limits of a function The concept of a limit of a function is one of the most important in mathematics. Many key concepts are defined in terms of limits (e.g., derivatives and continuity). The primary objective of this section is to help you acquire a firm intuitive understanding of the concept. Roughly speaking, the limit concept is concerned with the behavior of a function f around a certain point (say a ) rather than with the value fa() of the function at that point. The question is: what happens to the value of fx() when x gets closer and closer to a (without ever reaching a )? ln x Example 7.1 Take the function fx()= . 5 x2 −1 4 This function is not defined at the point x =1, because = 3 the denominator at x 1 is zero. However, as x f(x) = ln(x) / (x2-1) 1 2 ‘tends’ to 1, the value of the function ‘tends’ to 2 . We 1 say that “the limit of the function fx() is 2 as x 1 approaches 1, or 0 0 0.5 1 1.5 2 lnx 1 lim = x→1 x2 −1 2 It is important to be clear: the limit of a function and the value of a function are two completely different concepts. In example 7.1, for instance, the value of fx() at x =1 does not even exist; the function is undefined there. However, the limit as x →1 does exist: the value of the function fx() does tend to 1 something (in this case: 2 ) as x gets closer and closer to 1.
    [Show full text]
  • Computer Problems for Vector Calculus
    Chapter: Vector Calculus Computer Problems for Vector Calculus 2 2 1. The average rainfall in Flobbertown follows the strange pattern R = (1 + sin x) e−x y , where x and y are distances north and east from one corner of the town. (a) Pick at least 2 values of x and sketch the rainfall function R(y) at that value. Label your plots with their x values. (b) Pick at least 2 values of y and sketch the rainfall function R(x) at that value. Label your plots with their x values. (c) Have a computer generate a 3D plot of R(x; y). Make sure you plot over a region that clearly shows the general behavior of the function, and includes all the x and y values you used for your sketches. Check if the computer plot matches your constant-latitude and constant-longitude sketches. (If it doesn't, figure out what you did wrong.) (d) Based on your computer plot, if you were to start at the position (π=4; 1) roughly what direction could you move in to keep R constant? Hint: You may find it easier to answer this if you make a second plot that zooms in on a small region around this point. 2. The town of Chewandswallow has been buried in piles of bread. The depth of bread is given by B = cos(x + y) + sin x2 + y2, where the town covers the region 0 ≤ x ≤ π=2, 0 ≤ y ≤ π. (a) Have a computer make a contour plot of B(x; y).
    [Show full text]