Lecture07

# Lecture07 - Lecture 7 Interlude Uniqueness of the...

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Unformatted text preview: Lecture 7 Interlude : Uniqueness of the derivative. There is at most one A ∈ L ( X,Y ) such that F ( a + h )- F ( a )- A ( h ) k h k → as h → Proof 1. Lecture 4 notes Proof 2. Suppose A : X → Y is linear and satisfies lim h → F ( a + h )- F ( a )- A ( h ) k h k = 0 . Fix a,h . Introduce t ∈ R . Then as t → 0, 1 k th k ( F ( a + th )- F ( a )- A ( th )) → and 1 k th k k F ( a + th )- F ( a )- A ( th ) k = 1 | t |k h k k F ( a + th )- F ( a )- tA ( h ) k = 1 k h k F ( a + th )- F ( a ) t- A ( h ) so A ( h ) = lim t → F ( a + th )- F ( a ) t . So F a : X → Y is given by F a ( h ) = lim t → F ( a + th )- F ( a ) t = d dt F ( a + th ) | t =0 . 1 Chain Rule Examples Example : If F : U → Y and G : U → Y are differentiable at a ∈ U then so is F + G : U → Y and ( F + G ) a = F a + G a . Proof. We will use chain rule, though this can of course be proved directly. We can write F + G = S ◦ ( F,G ) where ( F,G ): U → Y × Y : x 7→ ( F ( x ) ,G ( x )) and S : Y ×...
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Lecture07 - Lecture 7 Interlude Uniqueness of the...

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