2433-Ch15-slides - THE GRADIENT Given a function of several...

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THE GRADIENT Given a function of several variables: z = f ( x,y )o r w = F ( x,y,z ) The gradient of f is the vector: f = ∂f ∂x i + ∂y j = f x i + f y j The gradient of F is the vector: F = ∂F i + j + ∂z k = F x i + F y j + F z k 1
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I. Directional Derivatives Given a function z = f ( x,y ) and a unit vector u = u 1 i + u 2 j . The directional derivative of f at the point ( x 0 ,y 0 ) in the direction u is the number f ± u ( x 0 0 )= f ( x 0 0 ) · u = f x ( x 0 0 ) u 1 + f y ( x 0 0 ) u 2 Geometric interpretation: 2
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Given a function w = F ( x,y,z ) and a unit vector u = u 1 i + u 2 j + u 3 k . The directional derivative of F at the point P 0 ( x 0 ,y 0 ,z 0 ) in the di- rection u is the number F ± u ( x 0 0 0 )= F ( x 0 0 0 ) · u = F x ( P 0 ) u 1 + F y ( P 0 ) u 2 + F z ( P 0 ) u 3 3
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Examples 1. f ( x,y )=2 x 2 y 3 - 3 y x (a) Find the directional derivative of f at the point (1 , - 1) in the direction of the vector a =3 i +4 j . (b) Find the directional derivative of f at the point (1 , - 1) in the direction of the line y = 5 2 x +2 4
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2. F ( x,y,z )= xy 2 + yz 2 + zx 2 Find the directional derivative of F at the point (1 , - 1 , 2) toward the point (2 , 3 , 4). 5
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II. Maximum and minimum di- rectional derivatives Functions of two variables: z = f ( x,y ) f ± u = f ( x 0 ,y 0 ) · u = ±∇ f ± cos θ where θ is the angle between f and u . max. direc. deriv. = ±∇ f ( x 0 0 ) ± min. direc. deriv. = -±∇ f ( x 0 0 ) ± 6
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Functions of three variables: w = F ( x,y,z ) F ± u = F ( x 0 ,y 0 ,z 0 ) · u = ±∇ F ± cos θ where θ is the angle between F and u . max. direc. deriv. = ±∇ F ( x 0 0 0 ) ± min. direc. deriv. = -±∇ F ( x 0 0 0 ) ± 7
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Examples: 1. f ( x,y )=2 x - x 2 y +2 y 2 - x y . (a) Find the maximum directional derivative of f at the point (2 , 1). (b) Find a unit vector in the di- rection of the minimum directional derivative of f at the point (1 , - 1). 8
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2. F ( x,y,z )=2 x 2 +3 xyz - yz 2 . (a) Find the minimum directional derivative of F at the point (2 , - 1 , 2). (b) Find a unit vector in the di- rection of the maximum directional derivative of f at the point (2 , - 1 , 2). 9
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III. Chain rules: Functions of 2 variables: z = f ( x,y ). If x = x ( t ) and y = y ( t ), then the composition f [ x ( t ) ,y ( t )] is a function of t and df dt = ∂f ∂x dx dt + ∂y dy dt = f · r ± ( t ) . 10
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If x = x ( s,t ) and y = y ( ), then the composition f [ x ( ) ,y ( )] is a function of s and t and ∂f ∂s = ∂x + ∂y . ∂t = + . 11
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Examples: 1. Set f ( x,y )=2 x 2 y + y 3 , where x ( t )= t 2 +1 ,y ( t e 2 t . F ( t f [ x ( t ) ( t )] = 2( t 2 +1) 2 e 2 t + e 6 t Find dF dt 12
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2. Set f ( x,y )=2 x 2 y + y 3 , where x ( s,t )= s 2 t, y ( te 2 s . F [( )] = f [ x ( ) ,y ( )] =2 s 4 t 3 e 2 s + t 3 e 6 s Find ∂F ∂s , ∂t 13
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Chain rules: Functions of 3 vari- ables w = F ( x,y,z ). If x = x ( t ) ,y = y ( t ) and z = z ( t ), then the composition F [ x ( t ) ( t ) ,z ( t )] is a function of t and dF dt = ∂F ∂x dx dt + ∂y dy dt + ∂z dz dt = F · r ± ( t ) . 14
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If x = x ( s,t ) ,y = y ( ) ,z = z ( ) , then the composition F [ x ( ) ( ) ( )] is a function of s and t and ∂F ∂s = ∂x + ∂y + ∂z . ∂t = + + . 15
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Examples: 1. Set F ( x,y,z )=2 x 3 y 2 - sin( xyz 2 ), where x ( t )=3 t 2 ,y ( t t +1 ,z ( t )= t 3 . F ( t F [ x ( t ) ( t ) ( t )] =54 t 6 (2 t +1) 2 - sin(6 t 6 +3 t 5 ) Find d F dt .
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2433-Ch15-slides - THE GRADIENT Given a function of several...

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