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.
 Fall '04
 unknown
 Calculus

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