H that is f a h f a bh when h is small theorem

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Unformatted text preview: , a map f : U → Rm , and a point a ∈ U , the function f is differentiable at a if there exists a linear mapping B : Rn → Rm such that for every h ∈ Rn − {0}, f (a + h) − f (a) − Bh → 0 as h → 0. |h| (2.15) That is, f (a + h) − f (a) ≈ Bh when h is small. Theorem 2.4. If f is differentiable at a, then for every u the directional derivative of f in the direction of u at a exists. Proof. The function f is differentiable at a, so f (a + tu) − f (a) − B (tu) → 0 as t → 0. |tu| (2.16) Furthermore, f (a + tu) − f (a) − B (tu) t f (a + tu) − f (a) − B (tu) = |tu| |tu| t � � t 1 f (a + tu) − f (a) − Bu = |t| |u| t → 0, (2.17) as t → 0, so f (a + tu) − f (a) → Bu as t → 0. t (2.18) Furthermore, the linear map B is unique, so the following definition is well­defined. Definition 2.5. The derivative of f at a is Df (a) = B , where B is the linear map defined above. Note that Df (a) : Rn → Rm is a linear map. 3 Theorem 2.6. If f is differentiable at a, then f is continuous at a. Ske...
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This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.

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