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

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ASSIGNMENT 5 · SOLUTIONS MAT 473 · SPRING 2012 Exercise 8.1. Suppose U R n is open and f, g : U R m are continuously di ff erentiable on U . Prove that the function ϕ : U R defined by ϕ ( x ) = f ( x ) , g ( x ) is continuously di ff erentiable on U . Proof. By the Product Rule (Proposition 3.8), ϕ is di ff erentiable on U , with ϕ ( x )( h ) = f ( x )( h ) , g ( x ) + f ( x ) , g ( x )( h ) for x U and h R n . Thus, for 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 ) , g ( y )( h ) . It follows from the Cauchy-Schwarz inequality (and the triangle inequality, and the operator- norm inequality) that if h 1, then ( ϕ ( x ) ϕ ( 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 ) g ( y ) h f ( x ) g ( x ) g ( y ) + f ( x ) g ( x ) g ( y ) + f ( x ) f ( y ) g ( y
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