Functions of Several Variables

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1 Functions of Several Variables Partial Derivatives Philippe B Laval KSU March 21, 2012 Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

2 Introduction In this section we extend the notion of derivative to functions of several variables We will emphasize the similarities as well as differences between functions of one variable and functions of several variables We begin with a quick review of the derivative for function of one variable Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

3 Quick Review You will recall that if y = f (x), then the derivative of f, denoted f (x) or dy dx or d dx f (x) is the instantaneous rate of change of f with respect to x We list some facts about the derivative students should know: 1 f f (x) f (a) f (a+h) f (a) (a) = lim x a x a = lim h 0 h 2 If f is differentiable at a point a ( f (a) exists) then f is continuous at a 3 f (a) is the slope of the tangent to y = f (x) at x = a 4 If f (x) > 0 on an interval, then f is increasing on that interval 5 If f (x) > 0 on an interval, then f is concave up on that interval ( f is increasing at an increasing rate) Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

4 General Ideas With functions of several variables, things get more difficult for the following reasons: If z = f (x, y), what do we mean by rate of change? If we think geometrically, the graph of z = f (x, y) is a surface in space What do we mean by slope? In this section, we will focus on rate of change with respect to x or y Later on, we will study rate of change with respect to both x and y Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

5 Partial Derivatives: Definition Consider the surface given by z = f (x, y) If we move on the surface by keeping y constant, say y = b, we are moving along the red curve The equation of the red curve is g (x) = f (x, b) Its slope is g g (x + h) g (x) f (x + h, b) f (x, b) (x) = lim = lim h 0 h h 0 h If we move on the surface by keeping x constant, say x = a, we are moving along the green curve The equation of the red curve is q (x) = f (a, y) Its slope is q q (y + h) q (y) f (a, y + h) f (a, y) (y) = lim = lim h 0 h h 0 h Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

6 Partial Derivatives: Definition y2 z x 2 4 Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

7 Partial Derivatives: Definition Using the idea above, we define the partial derivatives of f Definition Let f be a function of two variables 1 The partial derivative of f with respect to x, denoted f x (x, y) is defined to be: f x (x, y) = lim h 0 f (x + h, y) f (x, y) h 2 The partial derivative of f with respect to y, denoted f y (x, y) is defined to be: f y (x, y) = lim h 0 f (x, y + h) f (x, y) h Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

8 Partial Derivatives: Notation There are other notations for the partial derivatives Definition Let z = f (x, y) Then: 1 f x (x, y) = f x = f x = z x f (x, y) = x = f 1 = D 1 f = D x f 2 f y (x, y) = f y = f y = z y f (x, y) = y = f 2 = D 2 f = D y f The subscripts 1 and 2 represent the first and second variable which are x and y It becomes more relevant when we deal with functions having n variables, for large n Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

9 Partial Derivatives: Interpretation z y x Let (a, b) be the coordinates of the point at which the green and red curves intersect f x (a, b) is the slope of the red curve at (a, b, f (a, b)) f y (a, b) is the slope of the green curve at (a, b, f (a, b)) Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

10 Partial Derivatives: Computation Since partial derivatives are rates of change with respect to one variable only, we can use the rules of differentiation from Calculus I More specifically, we have: Rules to find partial derivatives of z = f (x, y) 1 To find f x, regard y as a constant and differentiate f (x, y) with respect to x 2 To find f y, regard x as a constant and differentiate f (x, y) with respect to y In the process outlined above, you can use all the rules of differentiation from Calculus I Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

11 Partial Derivatives: Examples Example Find f f x and y for f (x, y) = x 2 + y 2 + 5xy Example Find f f x and y for f (x, y) = ex2 +y Example Find z x if z is defined implicitly by x 2 + y 2 + z 2 = 4 Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

12 Higher Order Derivatives If f is a function in two variables, so are f x and f y So, we can differentiate them Their partial derivatives will also be functions in several variables, so we can differentiate them again Thus, we have the following: (f x ) x = f xx = f 11 = x (f y ) y = f yy = f 22 = y (f x ) y = f xy = f 12 = y (f y ) x = f yx = f 21 = x ( ) f x ( ) f y ( ) f x ( ) f y = 2 f x 2 = 2 z x 2 = 2 f y 2 = 2 z y 2 = 2 f y x = 2 z y x = 2 f x y = 2 z x y Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

13 Higher Order Derivatives Example Find the second order partial derivatives for z = f (x, y) = x 3 + y 3 + x 2 y 2 You will note that the mixed partials are the same This is not an accident Clairaut, a French mathematician ( ) proved the following theorem: Theorem (Clairaut s Theorem) Suppose that f is defined on a disk D which contains the point (a, b) If the functions f xy and f yx are both continuous on D, then f xy (a, b) = f yx (a, b) Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

14 Partial Differential Equations A partial differential equation is an equation which involves an unknown function and some of its partial derivatives Such equations arise in many applications in physics, chemistry, economics, We mention some of these equations here Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

15 Heat Equation ( u t = 2 ) k u x u y 2 u (x, y, t) gives the heat of an isotropic, homogeneous plate as a function of the position on the plate: (x, y) and time: t k is a constant which depends on the medium In 3 D, the heat equation becomes u t = k ( 2 u x u y u z 2 In this case u (x, y, z, t) gives the heat of an isotropic, homogeneous 3 D object as a function of the position on the object: (x, y, z) and time: t ) Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

16 Laplace Equation 2 u x u y 2 = 0 Solutions of this equation are called harmonic functions They play a role in problems related to heat conduction, fluid flow Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

17 Wave Equation 2 u t 2 = a2 2 u x 2 It describes the motion of a waveform Examples of waveforms include an ocean wave, a sound wave, a wave traveling along a vibrating string The equation written above corresponds to the vibration of a string u (x, t) gives the displacement of a string x units from one end of the string at time t In the case of an ocean wave, the function would be of the form u (x, y, t) and the wave equation would be 2 ( u t 2 = 2 ) u a2 x u y 2 The constant a is related to the vibrating medium Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

18 Partial Differential Equations: Examples Example Ṣhow that u (x, y) = e x sin y satisfies Laplace equation Example Ṣhow that u (x, t) = sin (x at) satisfies the wave equation Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

19 Exercises See the problems at the end of section 33 in my notes on partial derivatives Philippe B Laval (KSU) Functions of Several Variables March 21, / 19

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