Section11-1.6notes

Section11-1.6notes - 1 Section 11.6: Directional...

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Section 11.6: Directional Derivatives and the Gradient Vector Practice HW from Stewart Textbook (not to hand in) p. 778 # 1-4 p. 799 # 4-15, 17, 19, 21, 29, 35, 37 odd The Directional Derivative Recall that direction in the )) , ( , , ( point at the surface the to line tangent the of Slope ) , ( ) , ( x b a f b a x z b a f b a x = = direction in the )) , ( , , ( point at the surface the to line tangent the of Slope ) , ( ) , ( y b a f b a y z b a f b a y = = Instead of restricting ourselves to the x and y axis, suppose we want to find a method for finding the slope of the surface in any desired direction. 1 y ) 0 , , ( 0 0 y x z 0 x 0 y ) , ( y x f z = ) , , ( 0 0 0 z y x dir in Slope ) , ( 0 0 x y x f x = dir in Slope ) , ( 0 0 y y x f y = dir in Slope ) , ( 0 0 u y x D u = , = < b a u θ
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Let u = < a, b > be the unit vector (a vector of length one) on the x-y plane which indicates the direction we are moving. Then we define the following: Definition of the Directional Derivative The directional derivative of a function z = f ( x, y ) in the direction of the unit vector u = < a, b > , denoted by ) , ( y x f D u , is defined the be the following: b y x f a y x f y x f D y x ) , ( ) , ( ) , ( + = u Notes 1. Geometrically, the directional derivative is used to calculate the slope of the surface z = f ( x, y ). That is, to calculate the slope of the surface at the point ) , , ( 0 0 0 z y x , where ) , ( 0 0 0 y x f z = , we compute the following: b y x f a y x f y x f D b a vector unit of direction in z y x po at Surface of Slope y x ) , ( ) , ( ) , ( . ) , , ( int 0 0 0 0 0 0 0 0 0 + = = = < u u 2. The vector u = < a, b > must be a unit vector . If we want to compute the directional derivative of a function in the direction of the vector v and v is not a unit vector, we compute v v | v | v u | | 1 = = . 3. The direction of the unit vector u can be expressed in terms of the angle θ between the vector u and the x -axis. In this case, = < sin , cos u (note, u is a unit vector since 1 1 sin cos | | 2 2 = = + = u ) and the directional derivative can be expressed as sin ) , ( cos ) , ( ) , ( y x f y x f y x f D y x + = u . 4. Computationally, the directional derivative represents the rate of change of the function f in the direction of the unit vector u . 2
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Example 1: Find the directional derivative of the function x xy y y x f 6 4 3 ) , ( + - = at the point (1, 2) in the direction of the unit vector that makes an angle of 3 π θ = radians with the x -axis. Solution: 3
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Example 2: Find the directional derivative of the function x xy y y x f 6 4 3 ) , ( + - = at the point (-3, -4) in the direction of the vector j i v 3 2 + - = . Solution: 4
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Gradient of a Function Given a function of two variables z = f ( x, y ), the gradient vector, denoted by ) , ( y x f , is a vector in the x-y plane denoted by j i ) , ( ) , ( ) , ( y x f y x f y x f y x + = Facts about Gradients 1. The directional derivative of the function z = f ( x, y ) in the direction of the unit vector u = < a, b > can be expressed in terms of gradient using the dot product. That is, b y x f a y x f b a y
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This note was uploaded on 01/15/2010 for the course MATH MATH 262 taught by Professor Gregrix during the Spring '09 term at McGill.

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Section11-1.6notes - 1 Section 11.6: Directional...

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