fyG4o5-05

fyG4o5-05 - ASSIGNMENT 5 SOLUTIONS MAT 473 SPRING 2012...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
ASSIGNMENT 5 · SOLUTIONS MAT 473 · SPRING 2012 Exercise 8.1. Suppose U R n is open and f,g : U R m are continuously diferentiable on U .P rov etha ttheFunc t ion ϕ : U R de±ned by ϕ ( x )= ° f ( x ) ,g ( x ) ± is continuously diferentiable on U . Proof. By the Product Rule (Proposition 3.8), ϕ is diferentiable on U ,w ith ϕ ° ( x )( h )= ° f ° ( x )( h ) ,g ( x ) ± + ° f ( x ) ,g ° ( x )( h ) ± For x U and h R n .Thu s ,Fo r x , y U and h R n we have ( ϕ ° ( x ) ϕ ° ( y ))( h ) = ϕ ° ( x )( h ) ϕ ° ( y )( h ) = ° f ° ( x )( h ) ,g ( x ) ± + ° f ( x ) ,g ° ( x )( h ) ±−° f ° ( y )( h ) ,g ( y ) ±−° f ( y ) ,g ° ( y )( h ) ± = ° f ° ( x )( h ) ,g ( x ) ±−° f ° ( x )( h ) ,g ( y ) ± + ° f ( x ) ,g ° ( x )( h ) ±−° f ( x ) ,g ° ( y )( h ) ± −° f ° ( y )( h ) ,g ( y ) ± + ° f ° ( x )( h ) ,g ( y ) ±−° f ( y ) ,g ° ( y )( h ) ± + ° f ( x ) ,g ° ( y )( h ) ± = ° f ° ( x )( h ) ,g ( x ) g ( y ) ± + ° f ( x ) ,g ° ( x )( h ) g ° ( y )( h ) ± + ° f ° ( x )( h ) f ° ( y )( h ) ,g ( y ) ± + ° f ( x ) f ( y ) ,g ° ( y )( h ) ± = ° f ° ( x )( h ) ,g ( x ) g ( y ) ± + ° f ( x ) , ( g ° ( x ) g ° ( y ))( h ) ± + ° ( f ° ( x ) f ° ( y ))( h ) ,g ( y ) ± + ° f ( x ) f ( y )
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 03/11/2012.

Page1 / 2

fyG4o5-05 - ASSIGNMENT 5 SOLUTIONS MAT 473 SPRING 2012...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online