From Math 2220 Class 6 Tangent Lines Planes in R 3 Lines in R 3 Dr. - PowerPoint PPT Presentation

From Math 2220 Class 6 V1c Chain Rule Tangent Planes From Math 2220 Class 6 Tangent Lines Planes in R 3 Lines in R 3 Dr. Allen Back Cross product Sep. 10, 2014

Chain Rule The chain rule in multivariable calculus is in some ways very From Math 2220 Class 6 simple. But it can lead to extremely intricate sorts of V1c relationships (try thermodynamics in physical chemistry . . . ) as Chain Rule well as counter-intuitive looking formulas like Tangent Planes ∂ z ∂ y Tangent Lines ∂ x ∂ x = – . Planes in R 3 ∂ z ∂ y Lines in R 3 Cross product (The above in a context where f ( x , y , z ) = C . )

Chain Rule From Math 2220 Class 6 V1c First let’s try the conceptually simple point of view, using the Chain Rule fact that derivatives of functions are linear transformations. Tangent Planes (Matrices.) Tangent Lines Planes in R 3 Lines in R 3 Cross product

Chain Rule Think about differentiable functions From Math 2220 Class 6 g : U ⊂ R n → R m V1c Chain Rule and Tangent Planes f : V ⊂ R m → R p Tangent Lines Planes in R 3 where the image of f, namely ( f ( U )) is a subset of the domain Lines in R 3 V of g . Cross product The chain rule is about the derivative of the composition f ◦ g .

Chain Rule Here’s a picture: From Math 2220 Class 6 V1c Chain Rule Tangent Planes Tangent Lines Planes in R 3 Lines in R 3 Cross product

Chain Rule For g : U ⊂ R n → R m and f : V ⊂ R m → R p , let’s use From Math 2220 Class 6 V1c R n to denote a point of p R m Chain Rule q to denote a point of R p . Tangent r to denote a point of Planes Tangent Lines So more colloquially, we might write Planes in R 3 q = g ( p ) Lines in R 3 Cross product r = f ( q ) and so of course f ◦ g gives the relationship r = f ( g ( p )) . (The latter is ( f ◦ g )( p ) . )

Chain Rule From Math 2220 Class 6 V1c Chain Rule Tangent Planes Tangent Lines Planes in R 3 Lines in R 3 Cross product

Chain Rule From Math 2220 Class 6 V1c Fix a point p 0 with g ( p 0 ) = q 0 and f ( q 0 ) = r 0 . Let the Chain Rule derivatives of g and f at the relevant points be Tangent Planes Tangent Lines T = Dg ( p 0 ) S = Df ( q 0 ) . Planes in R 3 Lines in R 3 How are the changes in p , q , and r related? Cross product

Chain Rule From Math 2220 Class 6 Fix a point p 0 with g ( p 0 ) = q 0 and f ( q 0 ) = r 0 . Let the V1c derivatives of g and f at the relevant points be Chain Rule T = Dg ( p 0 ) S = Df ( q 0 ) . Tangent Planes How are the changes in p , q , and r related? Tangent Lines By the linear approximation properties of the derivative, Planes in R 3 Lines in R 3 ∆ q ∼ T ∆ p ∆ r ∼ S ∆ q Cross product And so plugging the first approximate equality into the second gives the approximation ∆ r ∼ S ( T ∆ p ) = ( ST )∆ p .

Chain Rule From Math 2220 Class 6 V1c Chain Rule Tangent ∆ r ∼ ( ST )∆ p . Planes Tangent Lines Planes in R 3 What is this saying? Lines in R 3 Cross product

Chain Rule From Math 2220 Class 6 V1c ∆ r ∼ ( ST )∆ p . Chain Rule Tangent Planes What is this saying? For g : U ⊂ R n → R m and f : V ⊂ R m → R p , Tangent Lines Planes in R 3 T = Df ( p 0 ) is an m × n matrix Lines in R 3 Cross product S = Dg ( q 0 ) is an p × m matrix So the product ST is a p × n matrix representing the derivative at p 0 of g ◦ f .

Chain Rule From Math 2220 Class 6 ∆ q = T ∆ p ∆ r = S ∆ q ∆ r = ST ∆ p V1c Chain Rule Tangent Planes Tangent Lines Planes in R 3 Lines in R 3 Cross product

Chain Rule From Math 2220 Class 6 V1c Chain Rule So the chain rule theorem says that if f is differentiable at p 0 Tangent with f ( p 0 ) = q 0 and g is differentiable at q 0 , then g ◦ f is also Planes differentiable at p 0 with derivative the matrix product Tangent Lines Planes in R 3 ( Dg ( q 0 )) ( Df ( p 0 )) . Lines in R 3 Cross product

Chain Rule From Math 2220 Class 6 V1c Chain Rule Problem: Suppose we have the polar coordinate map Tangent Planes g ( r , θ ) = ( r cos θ, r sin θ ) Tangent Lines Planes in R 3 Lines in R 3 and ( r , θ ) = f ( u , v ) is given by f ( u , v ) = ( uv , v ) . Find the Cross product derivative of g ◦ f .

Chain Rule From Math 2220 Class 6 V1c Chain Rule Tangent Planes f : U ⊂ R 2 → R and g : R → R 2 . Derivatives/Partial Tangent Lines derivatives of f ◦ g and g ◦ f ? Planes in R 3 Lines in R 3 Cross product

Chain Rule From Math 2220 Class 6 V1c e.g. Chain Rule z = z ( x , y ) x = x ( t ) y = y ( t ) Tangent Planes or explicitly Tangent Lines x 2 + y 2 � Planes in R 3 z = x = cos t y = 2 sin t Lines in R 3 Cross product f : R 2 → R c : R → R 2 c ( t ) =( x ( t ) , y ( t )) f ◦ c : R → R c ◦ f : R 2 → R 2

Chain Rule From Math 2220 Class 6 V1c Chain Rule Tangent c ( t ) = ( x ( t ) , y ( t )) x = cos t y = 2 sin t Planes Tangent Lines Planes in R 3 (A vector valued function with 1 dimensional domain is Lines in R 3 sometimes interpreted as a path c . It’s image is a curve ; the above c ( t ) could parametrize the ellipse 4 x 2 + y 2 = 4 . ) Cross product

Chain Rule From Math 2220 Class 6 V1c � � Chain Rule ∂ f ∂ f Df = ∂ x ∂ y Tangent � dx Planes � dt Dc = Tangent Lines dy Planes in R 3 dt � � dx Lines in R 3 � � ∂ f ∂ f D ( f ◦ c ) = dt ∂ x ∂ y dy Cross product dt ∂ f dx dt + ∂ f dy = ∂ x ∂ y dt

Chain Rule From Math 2220 Class 6 V1c � � Chain Rule ∂ f ∂ f Df = ∂ x ∂ y Tangent � dx Planes � dt Dc = Tangent Lines dy Planes in R 3 dt � � dx Lines in R 3 � � ∂ f ∂ f D ( f ◦ c ) = dt ∂ x ∂ y dy Cross product dt ∂ f dx dt + ∂ f dy = ∂ x ∂ y dt

Chain Rule From Math 2220 Class 6 V1c � � Chain Rule ∂ f ∂ f Df = ∂ x ∂ y Tangent � dx Planes � dt Dc = Tangent Lines dy Planes in R 3 dt � dx Lines in R 3 � � � ∂ f ∂ f D ( c ◦ f ) = dt dy ∂ x ∂ y Cross product dt � dx � ∂ f dx ∂ f dt ∂ x dt ∂ y = dy dy ∂ f ∂ f dt ∂ x dt ∂ y

Chain Rule From Math 2220 Class 6 V1c If we use t to denote both scalars in the domain of c and the Chain Rule range of f (instead of z for the latter), the above might more Tangent Planes intuitively be written as Tangent Lines Planes in R 3 � dx � ∂ t dx ∂ t Lines in R 3 dt ∂ x dt ∂ y D ( c ◦ f ) = dy dy ∂ t ∂ t Cross product dt ∂ x dt ∂ y

Chain Rule From Math 2220 Class 6 V1c If we use t to denote both scalars in the domain of c and the Chain Rule range of f (instead of z for the latter), the above might more Tangent intuitively be written as Planes � dx Tangent Lines � ∂ t dx ∂ t dt ∂ x dt ∂ y Planes in R 3 D ( c ◦ f ) = dy ∂ t dy ∂ t Lines in R 3 ∂ x ∂ y dt dt Cross product where more confusingly, using t = t ( x , y ) instead of z = f ( x , y ) we have c ( f ( x , y )) = ( x ( t ( x , y ) , y ( t ( x , y )) .

Chain Rule From Math 2220 Class 6 V1c Chain Rule Alternatively Tangent Planes z = f ( x 1 , x 2 ) t = ( c 1 ( z ) , c 2 ( z )) t = ( c 1 ( f ( x 1 , x 2 )) , c 2 ( f ( x 1 , x 2 ))) Tangent Lines Planes in R 3 looks quite sensible. Lines in R 3 Tradeoffs among naturality, intuitiveness, and precision are why Cross product we have so many notations for derivatives.

Chain Rule From Math 2220 Class 6 V1c Tree diagrams can be helpful in showing the dependencies for Chain Rule chain rule applications: Tangent Planes z = z ( x , y ) Tangent Lines x = x ( t ) Planes in R 3 y = y ( t ) Lines in R 3 Cross product z = z ( x ( t ) , y ( t )) .

Chain Rule From Math 2220 Class 6 V1c Chain Rule z = z ( x ( t ) , y ( t )) Tangent Planes Tangent Lines Planes in R 3 Lines in R 3 Cross product ∂ z dx dt + ∂ z dy ∂ x ∂ y dt

Chain Rule From Math 2220 Class 6 V1c Chain Rule z = z ( x ( t ) , y ( t )) Tangent Planes Intuitively, one might think: Tangent Lines 1 A change ∆ t in t causes a change ∆ x in x with multiplier Planes in R 3 dx dt . Lines in R 3 2 The change ∆ x in x contributes to a further change ∆ z in Cross product z with multiplier ∂ z ∂ x . So the overall contribution to the change in z from the x part has multiplier ∂ z dx ∂ x dt times ∆ t . 3 Similarly for the y part.

Chain Rule From Math 2220 Class 6 V1c Chain Rule z = z ( x , y ) Tangent x = x ( u , v ) Planes Tangent Lines y = y ( u , v ) Planes in R 3 z = z ( x ( u , v ) , y ( u , v )) . Lines in R 3 Cross product

Chain Rule From Math 2220 Class 6 V1c Chain Rule z = z ( x ( u , v ) , y ( u , v )) . Tangent Planes Tangent Lines Planes in R 3 Lines in R 3 Cross product ∂ z ∂ u = ∂ z ∂ x ∂ u + ∂ z ∂ y ∂ u . ∂ x ∂ y

Chain Rule From Math 2220 Class 6 V1c Problem: e x 2 y Chain Rule z = Tangent w = cos ( x + y ) Planes u 2 − v 2 Tangent Lines x = Planes in R 3 y = 2 uv Lines in R 3 Cross product ∂ z ∂ u ?

Chain Rule From Math 2220 Class 6 V1c Chain Rule Tangent Planes Cases like z = f ( x , u ( x , y ) , v ( y )) . Tangent Lines Planes in R 3 Lines in R 3 Cross product

Chain Rule From Math 2220 Class 6 V1c More formal approach: Chain Rule f : R 3 → R Tangent Planes u : R 2 → R Tangent Lines v : R → R Planes in R 3 h : R 2 → R Lines in R 3 Cross product h ( x , y ) = f ( x , u ( x , y ) , v ( y )) Dh =?