# quiz2-soln - MAT237Y – Multivariable Calculus Summer 2009...

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Unformatted text preview: MAT237Y – Multivariable Calculus Summer 2009 Quiz #2 SOLUTIONS Tuesday, July 9 from 6:10pm to 7:00pm. Instructors: A. Hammerlindl and J. Uren 1. (a) [3 marks] State the definition of the Fr´ echet derivative. Solution: For a point a , if there is a linear map T : R n → R m such that lim x → 1 bardbl x bardbl bardbl f ( x + a ) − f ( a ) − T ( x ) bardbl = 0 then f is differentiable at a and T is the Fr´ echet derivative. (b) [7 marks] Show that if f : R n → R m and g : R n → R m are Fr´ echet differentiable at a ∈ R n , then the sum f + g : R n → R m , x mapsto→ f ( x ) + g ( x ) is Fr´ echet differentiable at a . Solution: Suppose that S, T ∈ L ( R n , R m ) are Fr´ echet derivatives of f and g respectively. Then 1 bardbl x bardbl bardbl ( f + g )( x + a ) − ( f + g )( a ) − ( S + T )( x ) bardbl = 1 bardbl x bardbl bardbl f ( x + a ) − f ( a ) − S ( x ) + g ( x + a ) − g ( a ) − T ( x ) bardbl ≤ 1 bardbl x bardbl bardbl f ( x + a ) − f ( a ) − S ( x ) bardbl + 1 bardbl x bardbl bardbl g ( x + a ) − g ( a ) − T ( x ) bardbl ....
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