14.6 Directional Derivatives and Gradient Vectors 1. Partial Derivates are nice, but they only tell us the rate of change of a function z = f x, y in the i and j direction. What if we are interested in the rate of change of the function z = f x, y in another direction? Suppose we want to find the rate of change of a function z = f x, y at the point in the direction of a unit vector u = a, b. Consider the surface S with equation z = f x, y. Let and define point P on S P x 0,z 0. Consider the vertical plane passing through P in the direction of u. Let C be the curve of intersection of that plane and S. The slope of the tangent line T to C at the point P is the rate of change of z in the direction of u Definition: The directional derivative of f at ( x 0 ) in the direction of a unit vector u = a, b is f ( x D if the limit exists. u f ( x 0 ) = lim 0 + ha + hb) f ( x 0 ) h 0 h Note: If u = 1,0 or u = 0,1, what do we get? z 0 = f ( x 0 ) ( x 0 ) Section 14.6 Page 1
Example 1: For the contour map for z = f x, y graphed below, (a) estimate the directional derivative of z in the northeast direction at the the point ( 4,3), (b) estimate the directional derivative of z in the direction of 1, 0 at the point 10, 0. 10 100 70 8 55 6 20 4 13 2 10 15 40 0 60 Theorem (3): If f is a differentiable function of x and y, then f has a directional derivative in the direction of any unit vector u = a, b and D u f x, y a + f y ( x, y)b = f x x, y Corollary: If the unit vector u = a, b makes an angle of θ with the positive x-axis, then we can write u = cosθ,sinθ, and Theorem 3 becomes: 0 2 4 6 8 10 Section 14.6 Page 2 90
Example 2: Find the directional derivative f x, y if and u is the unit vector given by angle θ = π / 4. What is f 2,1? z = x 2 y 3 y 4 2. The Gradient Vector: Notice that the directional derivative looks like the dot product of two vectors: The vector ( x, y), f y ( x, y) f x occurs in many other contexts beyond directional derivatives. " Thus, it gets a name, we call it the gradient of f, and is denoted grad f or f read del f. Definition: If f is a function of two variables x and y, then the gradient of f is the vector function f defined by = y2 f ( x, y) = f x ( x, y), f y ( x, y) = f x i + f f ( 1,2 ) Example 3: If f x, y, find f x, y and. x Note: From our above notation we can now express the directional derivative as a dot product: y j f ( x, y) = f x ( x, y)a + f y ( x, y)b Section 14.6 Page 3
Example 4: Let f x, y. Consider the point on the surface where and y = 1. (Or, think of f giving the temperature on a 2D plate at position x, y. (a) Write the directional derivative of f in the direction of v = 4i 3j as the dot product of two vectors at P o. (b) Compute the directional derivative from (a). (c) What is the meaning of our answer in part (b) as far as an ant on the surface is concerned. (d) At what angle from the xy-plane is an ant climbing or descending instantaneously as he starts walking in the direction of v = 4i 3j from P o. (e) Find the parametric equations of the tangent line to this surface at in the direction of v = 4i 3j. 3. Functions of Three Variables: We can apply similar methods and thought to functions in three variables, before, we interpret unit vector u. = 2x 3 y 4 3x 2 y 4 P o x = 3 f x, y,z. As as the rate of change of the function in the direction of the P o f x, y, z Section 14.6 Page 4
Definition (10): The directional derivative of f at u = a, b, c is if the limit exists. in the direction of a unit vector derivative becomes: If f x, y, z is differentiable and (a unit vector), then we can use the same method as before to show that: Defining the gradient vector of f in three variables the same as that in 2, then Or for short: Thus, we can write the direction derivative again as: (When first learning this, it is important to keep track where function values live, and where f Example 5: For f x, y,z, find the directional derivative of f at in the direction of v = 1, 2, 2. f ( x 0, z 0 ) = lim h 0 ( x 0,z o ) If time: We can use vector notation. We write x0 = x 0, y or x 0 0 = x 0,z 0. The equation " of the line through x 0 in the direction of u is given by. So the value of f at any point on the line can be expressed as f x 0 + hu. Then our definition of the directional u = a, b, c f ( x, y, z) = f x ( x, y, z)a + f y ( x, y, z)b + f z x, y, z f ( x, y,z) = f x ( x, y,z), f y ( x, y,z), f z ( x, y,z) f = f x, f y, f z = f x i + f y j + f z k f ( x 0, z 0 ) f x 0 + ha + hb, z 0 + hc f ( x, y, z) = f u c = xyz ( 3, 2, 6) h lives) Section 14.6 Page 5
4. Maximizing the Directional Derivative: If we have a function in 2 or 3 variables, we might want to know in what direction the functional value is increasing the fastest. Well, recall: Theorem (15): Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative f ( x ) is f ( x ) and it occurs when u has the same direction as the gradient vector f x. Example 6: If f x, y,z : (a) In what direction does f have a maximum rate of change at the point 1,1, 1? (b) What is the maximum rate of change at the point 1,1, 1? Example 7: The temperature at a point x, y, z is given by where T is measured in o C and x, y, and z are in meters. Find the rate of change of temperature at the point 2, 1, 2 in the direction toward the point. Example 8: An ant is standing on the surface defined by = ( x + y) / z f = f u = f u cosθ ( 3, 3,3) T ( x, y, z) = 200e x2 3y 2 9z 2 f ( x, y) = 2x 3 y 4 3x 2 y 4 at the point 3,1, 23 (same as example 4). (Or think ant on hot plate). (a) In what direction should the ant walk to increase altitude most rapidly (give your answer as a 2 vector). Section 14.6 Page 6
(b) What is his maximum rate of increase (at what angle to the xy-plane)? 5. Tangent Planes to Level Surfaces: Suppose S is a surface with equation, that is, it is a level surface of a function F of three variables. Let P x 0,z o be a point on S. Let C be any curve on S passing through point P. Then let C be described by a continuous vector function r ( t) = x( t), y( t), z( t) (this is a space curve). Let t be the value of t for which r 0 ( t) is at P. That is r ( t 0 ) = x( t 0 ), y( t 0 ), z( t 0 ) = x 0, z o. We know that F( x( t), y( t), z( t) ) = k. Why? So long as x( t), y( t), and z( t) are differentiable functions of t, so is F. Thus, we can differentiate both sides of F( x( t), y( t), z( t) ) = k with respect to t. The Chain Rule gives: Thus: The equation of the tangent plane to the level surface at the point x 0,z o is given by: = k F x, y, z F( x, y, z) = k F x ( x 0, z 0 )( x x 0 ) + F y ( x 0, z 0 )( y y 0 ) + F z ( x 0, z 0 )( z z 0 ) = 0 Section 14.6 Page 7
The normal line to S at P is the line passing through P and perpendicular to the tangent plane. So its direction is given by the of the tangent plane. If we consider the special case when S is the surface defined by z = f x, y : So the tangent plane equation becomes: This is what we saw in 14.4.2 So our generalization is consistent with what we said before Example 9: For the surface y = x 2 z 2, find the equations of: (a) the tangent plane, (b) the normal line, at the point 4, 7, 3. f x ( x 0 )( x x 0 ) + f y ( x 0 )( y y 0 ) ( z z 0 ) = 0 Section 14.6 Page 8