Differential Calculus of Several Variables David Perkinson Abstract. These are notes for a one semester course in the differential calculus of several variables. The first two chapters are a quick introduction to the derivative as the best affine approximation to a function at a point, calculated via the Jacobian matrix. Chapters 3 and 4 add the details and rigor. Chapter 5 is the basic theory of optimization: the gradient, the extreme value theorem, quadratic forms, the Hessian matrix, and Lagrange multipliers. Studying quadratic forms also gives an excuse for presenting Taylor’s theorem. Chapter 6 is an introduction to differential geometry. We start with a parametrization inducing a metric on its domain, but then show that a metric can be defined intrinsically via a first fundamental form. The chapter concludes with a discussion of geodesics. An appendix presents (without proof) three equivalent theorems: the inverse function theorem, the implicit function theorem, and a theorem about maps of constant rank. Version: January 31, 2008 Contents Chapter 1. Introduction 1 1. A first example 1 § 2. Calculating the derivative 6 § 3. Conclusion 13 § Chapter 2. Multivariate functions 17 1. Function basics 17 § 2. Interpreting functions 19 § 3. Conclusion 31 § Chapter 3. Linear algebra 35 1. Linear structure 35 § 2. Metric structure 37 § 3. Linear subspaces 43 § 4. Linear functions 49 § 5. Conclusion 57 § Chapter 4. The derivative 65 1. Introduction 65 § 2. Topology 65 § 3. Limits, continuity 68 § 4. The definition of the derivative 70 § 5. The best affine approximation revisited 72 § 6. The chain rule 73 § 7. Partial derivatives 77 § 8. ThederivativeisgivenbytheJacobianmatrix 79 § 9. Conclusion 83 § Chapter 5. Optimization 89 1. Introduction 89 § iii iv Contents 2. Directional derivatives and the gradient 89 § 3. Taylor’s theorem 92 § 4. Maxima and minima 97 § 5. The Hessian 106 § Chapter 6. Some Differential Geometry 115 1. Stretching 115 § 2. First Fundamental Form 116 § 3. Metrics 119 § 4. Lengths of curves 120 § 5. Geodesics 122 § Appendix A. Set notation 133 Appendix B. Real numbers 137 1. Field axioms 137 § 2. Order axioms 138 § 3. Least upper bound property 138 § 4. Interval notation 139 § Appendix C. Maps of Constant Rank 141 Appendix. Index 145 Chapter 1 Introduction The main point of differential calculus is to replace curvy things with flat things: to approx- imate complicated functions with linear functions. For example, in one variable calculus, one approximates the graph of a function using a tangent line: 4 2 -2 -1 1 2 0 x In the illustration above, the function g(x)= x2 is replaced by the simpler function ℓ(x)= 2x 1, a good approximation near the point x = 1. We begin these notes with an analogous − example from multivariable calculus. 1. A first example Consider the function f(u,v)=(u2 + v,uv). It takes a point (u,v) in the plane and hands back another point, (u2 + v,uv). For example, f(2, 3) = (22 + 3, 2 3) = (7, 6). · Our first task is to produce a picture of f. In one variable calculus, one usually pictures a function by drawing its graph. For example, one thinks of the function g(x) = x2 as a parabola by drawing points of the form (x,x2). If we try the same for our function, we 1 2 1. Introduction would need to draw points of the form (u,v,u2+v,uv), which would require four dimensions. Further, in some sense, the graph of f is the simplest geometric realization of f: it contains exactly the information necessary to picture f, and no more. Thus, as usual in multivariate calculus, we are forced to picture an object that naturally exists in a space of dimension higher than three. There are several ways of dealing with the problem of picturing objects involving too many dimensions, and in practice functions such as f arise in a context that suggests a particular approach. We will start with one important point of view. Suppose we want to picture f for u and v in the interval [0, 1]∗, so the points (u,v) lie in a unit square in the plane. Think of this unit square as a thin sheet of putty, and think of the function f as a set of instructions for stretching the putty into a new shape. For example, the point 2 1 1 (1, 1) is stretched out to the point f(1, 1) = (1 + 1, 1 1) = (2, 1); the point ( 2 , 2 ) moves 1 1 3 1 · to f( 2 , 2 )=( 4 , 4 ); the origin, (0, 0), remains fixed, f(0, 0) = (0, 0). The motion of these points and a few others is shown below. f (1,1) (2,1) (0,1/2) (1/2,1/2) (3/4,1/4) (1/2,0) (0,0) (1/2,0) (0,0) (1/4,0) Our goal is to see where every point in the unit square is moved by f. To start, consider what happens to the boundary of the square: bottom edge. The bottom edge of the square consists of points of the form (u, 0) as u varies from 0 to 1. Applying f gives f(u, 0)=(u2 + 0,u 0)=(u2, 0). · So as a point moves along the bottom edge at a constant unit speed from (0, 0) to (1, 0), its image under f moves between the same two points, moving slowly at first, then more and more quickly, (velocity = 2u). right edge. The right edge consists of points of the form (1,v), and f(1,v) = (1+ v,v); the image is a line segment starting at (1, 0) when v = 0 and ending at (2, 1) when v = 1. top edge. Points along the top have the form (u, 1), and f(u, 1) = (u2 + 1,u). Calling the first and second coordinates x and y, we have x = u2 +1 and y = u. Thus, points in the image of the top edge satisfy the equation x = y2 + 1, forming a parabolic arc from (2, 1) to (1, 0) as we travel from right to left along the top edge of the square. ∗[0, 1] denotes the real numbers between 0 and 1, including the endpoints. For a reminder on interval notation, see Appendix B. 1. A first example 3 left edge. Points on the left edge have the form (0,v), and f(0,v) = (v, 0). Thus, we can picture f as flipping the left edge over a 45◦ line, placing it along the bottom edge. The following picture summarizes what f does to the boundary of the unit square: 3 3 f 4 2 2 1 1 4 We now want to figure out what is happening to the interior of the square. Considering what happens to the boundary, it seems that f is taking our unit square of putty, folding it along a diagonal, more or less, and lining up the left edge with the bottom edge. To get more information, draw a square grid of lines on the unit square, and plot the images of each of these lines under f. A vertical line, a distance c from the origin, will consist of points of the form (c,v) with v varying, and f will send these points to points of the form f(c,v)=(c2 + v,cv). The image is a line passing through (c2, 0) when v = 0, and (c2 + 1,c) when v = 1. A horizontal line at height c will consist of points of the form (u,c) which are sent by f to points of the form f(u,c)=(u2 + c,uc) lying on a parabolic arc connecting (c, 0) to (1+ c,c). The following picture shows the image of a grid under f: 1 1 0.8 0.6 0.5 f 0.4 0.2 0 0.5 1 0 0.5 1 1.5 2 You should be starting to see that multivariate functions are more interesting than the functions one meets in one variable calculus. Even though f involves only a few very simple terms, its geometry is fairly complicated. Differential calculus provides one main tool for dealing with this complexity: it shows how to approximate a function with a simpler type of function, namely, a linear function. We will give a rigorous definition of a linear function later (cf. page 49). For now, it is enough to know that we can easily analyze a linear function; there is no mystery to its geometry. The image of a square grid under a linear function is always a parallelogram; the image of a grid will be a grid of parallelograms. To be more precise, consider the function f. Given a point p in the unit square, differ- ential calculus will give us a linear function that closely approximates f provided we stay near the point p. (Given a different point, calculus will provide a different linear function.) 1 3 To illustrate the idea, take the point p = ( 4 , 4 ). The linear function provided by calculus turns out to be: 1 3 1 L(u,v)=( u + v, u + v). 2 4 4 4 1. Introduction (To spoil a secret, read the footnote.†) To picture L, consider the image of a unit square. The function L acts on the vertices 1 3 3 1 as follows: L(0, 0) = (0, 0), L(1, 0) = ( 2 , 4 ), L(1, 1) = ( 2 , 1), and L(0, 1) = (1, 4 ). It turns out that linear functions always send lines to lines; hence we can see how L acts on the boundary of the unit square: 3 2 L 4 2 1 3 4 1 The function L stretches, skews, and flips the unit square.