Real Analysis III (MAT312β) Department of Mathematics University of Ruhuna A.W.L. Pubudu Thilan Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 1/37 Chapter 5 Directional Derivatives and Partial Derivatives Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 2/37 Why do we need directional derivatives? Suppose y = f (x). Then the derivative f 0(x) is the rate at which y changes when we let x vary. Since f is a function on the real line, so the variable can only increase or decrease along that single line. In one dimension, there is only one "direction" in which x can change. Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 3/37 Why do we need directional derivatives? Cont... Given a function of two or more variables like z = f (x; y), there are infinitely many different directions from any point in which the function can change. We know that we can represent directions as vectors, particularly unit vectors when its only the direction and not the magnitude that concerns us. Directional derivatives are literally just derivatives or rates of change of a function in a particular direction. Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 4/37 Why do we need directional derivatives? Cont... Let P is a point in the domain of f (x; y) and vectors v1, v2, v3, and v4 represent possible directions in which we might want to know the rate of change of f (x; y). Suppose we may want to know the rate at which f (x; y) is changing along or in the direction of the vector, v3, which would be the direction along the x-axis. Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 5/37 Definition If y is a unit vector, then f (a + hy) − f (a) f 0(a; y) = lim ; h!0 h the derivative f 0(a; y) is called the directional derivative of f at a in the direction of y. Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 6/37 Directional derivative of f (x; y) at (a; b) in the direction of u If u = u1i + u2j is a unit vector, we define the directional derivative fu at the point (a; b) by fu(a; b) = Rate of change of f (x; y) in the direction of u at the point (a; b) f (a + hu ; b + hu ) − f (a; b) = lim 1 2 h!0 h provided that the limit exists. Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 7/37 Example 1 2 Compute the directional derivativep of f (x; y) = x + y at the point 1 3 (4, 0) in the direction u = i + j. 2 2 Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 8/37 Example 1 Solution r ( ) ( ) p 2 k k 1 2 3 The norm of u, that is u = 2 + 2 = 1. Thus u is a unit vector. f (a + hu1; b + hu2) − f (a; b) fu(4; 0) = lim h!0 h p f (4 + h 1 ; 0 + h 3 ) − f (4; 0) = lim 2 2 ! h 0 ( h ) ( ) p 2 1 3 − 4 + h 2 + h 2 4 = lim h!0 h 2 ( ) 4 + h + 3h − 4 1 3 1 = lim 2 4 = lim + h = h!0 h h!0 2 4 2 Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 9/37 Example 2 A scalar field f is defined on Rn by the equation f (x) = a:x, where a is a constant vector. Compute f 0(x; y) for arbitary x and y. Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 10/37 Example 2 Solution According to the definition, we have f (a + hy) − f (a) f 0(a; y) = lim h!0 h f (x + hy) − f (x) f 0(x; y) = lim h!0 h a:(x + hy) − a:x = lim h!0 h h(a:y) = lim h!0 h = a:y: Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 11/37 Example 3 Let T : Rn ! Rn be a given linear transformation. Compute the derivative f 0(x; y) for the scalar field defined on Rn by the equation f (x) = x:T (x). Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 12/37 Example 3 Solution According to the definition, we have f (a + hy) − f (a) f 0(a; y) = lim h!0 h f (x + hy) − f (x) f 0(x; y) = lim h!0 h (x + hy):T (x + hy) − x:T (x) = lim h!0 h (x + hy):(T (x) + hT (y)) − x:T (x) = lim h!0 h x:T (x) + hx:T (y) + hy:T (x) + h2y:T (y) − x:T (x) = lim h!0 h hx:T (y) + hy:T (x) + h2y:T (y) = lim h!0 h = x:T (y) + y:T (x): Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 13/37 Partial derivatives If y is a unit vector, the derivative f 0(a; y) is called the directional derivative of f at a in the direction of y. th In particular, if y = ek (the k unit coordinate vector) the 0 directional derivative f (a; ek ) is called partial derivative with respect to ek and is also denoted by the symbool Dk f (a). Thus f (a + hy) − f (a) f 0(a; y) = lim ; h!0 h 0 f (a; ek ) = Dk f (a): Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 14/37 Partial derivatives Notations The following notations are also used for the partial derivative Dk f (a): (i) Dk f (a1; :::; an), @f (ii) (a1; :::; an), @xk (iii) f 0 (a ; :::; a ). xk 1 n Sometimes the derivative f 0 is written without the prime as f or xk xk even more simply as fk . Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 15/37 Partial derivatives in R2 Notations In R2 the unit coordinate vectors are denoted by i and j. If a = (a; b) the partial derivatives f 0(a; i) and f 0(a; j) are also written as @f @f (a; b) and (a; b); @x @y respectively. Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 16/37 Partial derivatives in R3 Notations In R3 the unit coordinate vectors are denoted by i, j, and k. If a = (a; b; c) the partial derivatives D1f (a), D2f (a), and D3f (a) are denoted by @f @f @f (a; b; c); (a; b; c); and (a; b; c); @x @y @z respectively. Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 17/37 Partial derivatives of higher order Partial differentiation produces new scalar fileds D1f ; ::::; Dnf from a given scalar field f . The partial derivatives D1f ; ::::; Dnf are called first order partial derivatives of f . For function of two variables there are four second order partial derivatives, which are written as follows: @2f @2f D (D f ) = ; D (D f ) = 1 1 @x2 2 2 @y 2 @2f @2f D (D f ) = ; D (D f ) = : 1 2 @x@y 2 1 @y@x Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 18/37 Partial derivatives of higher order Cont... In the above, D1(D2f ) means the partial derivative of (D2f ) with respect to the first variable. We sometimes use the notation Di;j f for the second-order partial derivative Di (Dj f ). For example, D1;2f = D1(D2f ). Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 19/37 Partial derivatives of higher order Cont... In the @-notation we indicate the order of derivatives by writing ( ) @2f @ @f = : @x@y @x @y This may or may not be equal to the other mixed partial derivative, ( ) @2f @ @f = : @y@x @y @x Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 20/37 Partial derivatives of higher order Remark We shall prove later that the two mixed partials D1(D2f ) and D2(D1f ) are equal at a point if one of them is continuous in a neighborhood of the point. Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 21/37 Example Consider the function f (x; y) = x2 + 5xy − 4y 2: Find the second order partial derivatives of f . Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 22/37 Example Solution @f @f = 2x + 5y = 5x − 8y: @x @y A second order partial derivative should be a partial derivative of a first order partial derivative. So, first take two different first order partial derivatives, with respect to x or y and then, for each of those, you can take a partial derivative a second time with respect to x or y. Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 23/37 Example Solution)Cont... ( ) @2f @ @f @ D (D f ) = = = (2x + 5y) = 2; 1 1 @x2 @x @x @x ( ) @2f @ @f @ D (D f ) = = = (2x + 5y) = 5; 2 1 @y@x @y @x @y ( ) @2f @ @f @ D (D f ) = = = (5x − 8y) = 5; 1 2 @x@y @x @y @x ( ) @2f @ @f @ D (D f ) = = = (5x − 8y) = −8: 2 2 @y 2 @y @y @y Note that fxy and fyx are equal in this example. While this is not always the case. Department of Mathematics University of Ruhuna | Real Analysis III(MAT312β) 24/37 Directional derivatives and continuity Differentiable function in one dimensional space If a is a point in the domain of a function f , then f is said to be differentiable at a if the derivative f 0(a) exists: f (a + h) − f (a) f 0(a) = lim : h!0 h In calculus, a differentiable function is a function whose derivative exists at each point in its domain.