Advanced Calculus (Math 4305) A®Ë@ Õæ Aë áÖ ß @.X.@ 2015-2016 ÈðB@ É®Ë@ Chapter 2 Differential Calculus In this chapter we study the theory and applications of differentiable calculus for functions of several variables. 2.1 Differentiability in One Variable We begin with an approach to the notion of derivative that is a bit different from the one usually found in elementary calculus books. This point of view is very useful in more advanced work, and it is the one that leads to the proper notion of differentiability for functions of several variables. Definition. (Differentiability in one variable) Let f be a real-valued function defined on some open interval in R containing the point a. We say that f is differentiable at a if there is a number m such that E(h) f(a + h) = f(a) + mh + E(h); where lim = 0: h!0 h In other words, if f(a + h) is the sum of the linear function f(a) + mh and an error term that tends to zero more rapidly than h as h ! 0: Proposition 1. Let f be a real-valued function defined on some open interval in R containing the point a. Then f is differentiable at a if and only if f 0(a) exists. Proof. Remarks. (1) The above proposition shows that the new definition of differentiability is equivalent to the usual one; it simply puts more emphasis on the idea of linear approximation. (2) The number m in the new definition is uniquely determined, and it is the derivative of f at a (m = f 0(a)). Definition. (Little o notation) We say that a function f(x) is o(g(x)) (read: f is little oh of g) as x ! x0 if f(x) lim = 0: x!x0 g(x) E(h) Remark. It often convenient to express the condition lim = 0 in the definition of differentia- h!0 h bility by saying that E(h) is o(h), meaning that E(h) is of smaller order of magnitude than h. 1 Example 1. Prove that if f and g are differentiable at x = a, then fg is differentiable at x = a and (fg)0(a) = f 0(a)g(a) + f(a)g0(a): Solution. Definition. (One-sided Derivatives) (1) We define the right-hand derivative of a function f at a point a to be 0 f(a + h) − f(a) f+(a) = lim : h!0+ h (2) We define the left-hand derivative of a function f at a point a to be 0 f(a + h) − f(a) f−(a) = lim : h!0− h Remarks. (1) A function f is differentiable at a if and only if its right-hand and left-hand derivatives exist and are equal. (2) One-sided derivative are useful in discussing differentiability of functions at end points and functions whose graphs have corners. The Mean Value Theorem The definition of the derivative involves passing from the local information given by the values of f(x) for x near a to the infinitesimal information f 0(a), which (intuitively speaking) gives the infinitesimal change in f corresponding to an infinitesimal change in x. To reverse the process and pass from infinitesimal information to local information-that is, to extract information about f from a knowledge of f 0- the principal tool is the mean value theorem. The derivation begins with the following result, which is important in its own right. Proposition 2. Suppose that a function f is defined on an open interval I and a 2 I. If f has a local maximum or minimum at the point a and f is differentiable at a, then f 0(a) = 0. Proof. Exercise. (1) Prove the second case: f has a local maximum at a. (2) Prove that if f(x) ≥ 0 in some open interval containing x0 and lim f(x) = L, then L ≥ 0. x!x0 Lemma 1. (Rolle's Theorem) Suppose that f is continuous on [a; b] and differentiable on (a; b). If f(a) = f(b), then there is at least one point c 2 (a; b) such that f 0(c) = 0. Proof. Theorem 1. (Mean Value Theorem I) Suppose that f is continuous on [a; b] and differentiable on (a; b). Then there is at least one point f(b) − f(a) c 2 (a; b) such that f 0(c) = : b − a 2 Proof. Remark. The main application of Mean Value Theorem is the using of information about f 0 to deduce information about f. Definition. (Increasing and decreasing functions) (1) We say that a function f is increasing on an interval I if f(a) ≤ f(b) whenever a; b 2 I and a < b. (2) We say that a function f is strictly increasing on an interval I if f(a) < f(b) whenever a; b 2 I and a < b. (3) We say that a function f is decreasing on an interval I if f(a) ≥ f(b) whenever a; b 2 I and a < b. (4) We say that a function f is strictly decreasing on an interval I if f(a) > f(b) whenever a; b 2 I and a < b. Theorem 2. Suppose that f is differentiable on an open interval I. (1) If jf 0(x)j ≤ K for all x 2 I, then jf(b) − f(a)j ≤ Kjb − aj for all a; b 2 I. (2) If f 0(x) = 0 for all x 2 I, then f is constant on I. (3) If f 0(x) ≥ 0 (respectively f 0(x) > 0; f 0(x) ≤ 0, or f 0(x) < 0) for all x 2 I, then f is increasing (respectively strictly increasing, decreasing, or strictly decreasing) on I. Proof. Exercise. Prove the other cases in part (3). Theorem 3. (Mean Value Theorem II) Suppose that f and g are continuous on an interval [a; b] and differentiable on (a; b), and g0(x) 6= 0 for all x 2 (a; b). Then there exists c 2 (a; b) such that f 0(c) f(b) − f(a) = : g0(c) g(b) − g(a) Proof. Vector-Valued Functions n The differential calculus generalized easily to functions of a real variable with values in R : That n is, a function f : U ⊆ R ! R , f(x) = (f1(x); : : : ; fn(x)): Definition. (Derivative of Vector-Valued Functions) n Let f : U ⊆ R ! R be a vector-valued function. we define the derivative of f at a point a 2 U to be f(a + h) − f(a) f 0(a) = lim : h!0 h n Remarks. Let f : U ⊆ R ! R be a vector-valued function and let f(x) = (f1(x); : : : ; fn(x)): (1) lim f(x) = L = (L1;:::;Ln) if and only if lim fj(x) = Lj; j = 1; : : : ; n: x!a x!a 3 (2) A vector-valued function f = (f1; : : : ; fn) is differentiable at a point a if and only if each of its component functions fj is differentiable at a. Moreover, the differentiation can be performed componentwise: 0 0 0 f (a) = (f1(a); : : : ; fn(a)): Differentiation Rules Let f and g be vector-valued functions and let ' be a real-valued function. Then (1) ('f)0 = '0f + 'f 0. (2) (f · g)0 = f 0 · g + f · g0: (3) If n = 3, then (f × g)0 = f 0 × g + f × g0: Exercises (page 52): 1 − 9 2.2 Differentiability in Several Variables n In this section we will study differentiability of functions of several variables, f : U ⊆ R ! R: There are several notions of derivatives for functions of several variables. The simplest one is partial derivatives. Definition. (Partial Derivatives) The partial derivative of a function f(x1; : : : ; xn) with respect to the variable xj is @f f(x ; : : : ; x + h; : : : ; x ) − f(x ; : : : ; x ) = lim 1 j n 1 n ; @xj h!0 h provided that the limit exists. Notation. The partial derivatives of f with respect to xj is denoted by @f = fxj = fj = @xj f = @jf: @xj 8 x − y2 > ; x + y 6= 0 < x + y Example 1. Find f2(0; 0) if f(x; y) = > :> 0; x + y = 0: Solution. 1 1 Example 2. Find u and u if u(x; y) = e2y + e2x: x y x y Solution. Remark. The partial derivatives of a function give information about how the value of the function changes when just one of the independent variables changes; that is, they tell us how the function varies along the lines parallel to the coordinate axes. 4 Differentiability We need to give more thought to what it should mean for a function of several variables to be differentiable. The right idea is provided by the characterization of differentiability in one variable that we developed in the preceding section. Namely, a function f(x) is differentiable at a point x = a if there is a linear function l(x) such that l(a) = f(a) and the difference E(h) = f(x) − l(x) tends to zero faster than h = x − a as x approaches a. The general linear function of n variables has the form l(x) = b + c1x1 + c2x2 + ··· + cnxn = b + c · x: the condition l(a) = f(a) yields b = f(a) − c · a: Hence l(x) = f(a) + c · (x − a): Now, f(x) = l(x) + E(h) implies f(x) = f(a) + c · (x − a) + E(h) = f(a) + c · h + E(h): Definition. (Differentiability) n (1) A function f : S ⊆ R ! R is called differentiable at a point a = (a1; : : : ; an) 2 S if there is a n vector c = (c1; : : : ; cn) 2 R such that f(a + h) − f(a) − c · h lim = 0: h!0 jhj (2) If f is differentiable at a, then c is called the gradient of f at a and it is denoted by rf(a): Remark.