Math 241 Spring 2014 Math 241 Spring 2014 Jayadev S Athreya Spring 2014 Math 241 Spring 2014 Chain Rule 14 5 Derivatives along curves Suppose we have a parametric curve x t y t R2 and a function f R2 R We can make the function h t f x t y t How do we compute h0 t Math 241 Spring 2014 Chain Rule 14 5 One variable chain rule h t f x t then h0 t f 0 x t x 0 t Math 241 Spring 2014 Chain Rule 14 5 Chain rule h0 t fx x t y t x 0 t fy x t y t y 0 t Math 241 Spring 2014 Chain Rule 14 5 How do we get to this A linear approximation to f near a b can be written as f fx a b x fy a b y f f x y f a b x x a y y b Math 241 Spring 2014 Chain Rule 14 5 Approximating h h t t h t h0 t t h t t f x t t y t t x t t x t x 0 t t y t t y t y 0 t t Math 241 Spring 2014 Chain Rule 14 5 Combining Approximations h t t f x t t y t t f x t x 0 t t y t y 0 t t Math 241 Spring 2014 Chain Rule 14 5 Combining Approximations h t t f x t t y t t f x t x 0 t t y t y 0 t t h t t h t f x t x 0 t t y t y 0 t t f x t y t Math 241 Spring 2014 Chain Rule 14 5 Combining Approximations h t t f x t t y t t f x t x 0 t t y t y 0 t t h t t h t f x t x 0 t t y t y 0 t t f x t y t fx x t y t x 0 t t fy x t y t y 0 t t Math 241 Spring 2014 Chain Rule 14 5 Combining Approximations h t t f x t t y t t f x t x 0 t t y t y 0 t t h t t h t f x t x 0 t t y t y 0 t t f x t y t fx x t y t x 0 t t fy x t y t y 0 t t t fx x t y t x 0 t fy x t y t y 0 t Math 241 Spring 2014 Chain Rule 14 5 Notation z f x y x x t y y t z dx z dy dz dt x dt y dt Math 241 Spring 2014 Chain Rule 14 5 Example Math 241 Spring 2014 Chain Rule 14 5 Higher dimensions Chain rule works in any dimension f Rn R x1 xn R R h t f x1 t xn t f x t Math 241 Spring 2014 Chain Rule 14 5 Higher dimensions Chain rule works in any dimension f Rn R x1 xn R R h t f x1 t xn t f x t h0 t fx1 x t x10 t fxn x t xn0 t Math 241 Spring 2014 Chain Rule 14 5 Higher dimensional inputs x x u v y y u v z z x y z x u v y u v Then z x z y z u x u y u z z x z y v x v y v Partial derivatives reduce this to case of 1 variable inputs Math 241 Spring 2014 Chain Rule 14 5 Geometric example z z x y x 2 y 2 x x r r cos y y r r sin Math 241 Spring 2014 Chain Rule 14 5 Geometric example z z x y x 2 y 2 x x r r cos y y r r sin z Math 241 Spring 2014 Chain Rule 14 5 Geometric example z z x y x 2 y 2 x x r r cos y y r r sin z x z y z 2x r sin 2y r cos x y 2r cos r sin 2r sin r cos 0 Math 241 Spring 2014 Chain Rule 14 5 Geometric example z z x y x 2 y 2 x x r r cos y y r r sin Math 241 Spring 2014 Chain Rule 14 5 Geometric example z z x y x 2 y 2 x x r r cos y y r r sin z r Math 241 Spring 2014 Chain Rule 14 5 Geometric example z z x y x 2 y 2 x x r r cos y y r r sin z x z y z 2x cos 2y sin x r y r r 2r cos cos 2r sin sin 2r sin2 cos2 2r Math 241 Spring 2014 Directional Derivatives 14 6 Other directions Partial derivatives are rates of change along lines parallel to the x and y axes What about other lines Math 241 Spring 2014 Directional Derivatives 14 6 Lines through a b A line through a b in direction v v1 v2 is given by x t a v1 t y t b v2 t h t f x t y t Dv f a b h0 0 Math 241 Spring 2014 Directional Derivatives 14 6 Applying Chain Rule x t a v1 t y t b v2 t h t f x t y t h0 0 fx a b x 0 0 fy a b y 0 0 v1 fx a b v2 fy a b Dv f a b v fx a b fy a b Math 241 Spring 2014 Gradient 14 6 The Gradient Given f R2 R we define the gradient f of f by f a b fx a b fy a b fx a b i fy a b j Math 241 Spring 2014 Gradient 14 6 The Gradient Given f R2 R we define the gradient f of f by f a b fx a b fy a b fx a b i fy a b j Dv f a b v f a b Math 241 Spring 2014 Gradient 14 6 Three dimensions For f R3 R f a b c fx a b c fy a b c fz a b c fx a b c i fy a b c j fz a b c k Math 241 Spring 2014 Gradient 14 6 Three dimensions For f R3 R f a b c fx a b c fy a b c fz a b c fx a b c i fy a b c j fz a b c k For v R3 Dv f a b c v f a b c Math 241 Spring 2014 Gradient 14 6 n dimensions For f Rn R f x fx1 x fx2 x fxn x …
View Full Document
Unlocking...