#### Objective

1. know the difference between the derivative of a vector function and the derivative of a function defined on an open set of a vector normed space.
2. Know the relation between £ (E×F, G) and £(E, £ (F, G))
3. Know the mean-value theorem with all its applications for the cases :
i- ƒ : [a, b] E where E is a vector normed space and [a, b] R.
ii- ƒ : [a, b] F where E and F are two vector normed spaces and [a, b] U is an open set of E.
iii- ƒ : U F where E and F are two vector normed spaces and U is an open connected set of E.
4. Know the primitive of a regular function and the derivation under the sum signe.
5. Know the local inversion theorem and implicit functions theorem.
6. Know the notion of derivative of superior order of a function ƒ defined on an open set and Taylor formulas.
7. Know the necessary conditions and one sufficient condition for local extremum of a function ƒ : U R where U is an open set of a vector normed space.

#### Content

• Derivative of vector functions – Mean-value theorem for these functions.
• Complements for the vector normed spaces £(E) where E is a Banah space, £(E,F) et Isom(E,F) if E and F are Banach spaces and the space of multilinear mappings £n(E,F). Study of the mapping : Isom(E, F) £(F, E) defined by (u)= u-1.
• Differentiable functions from an open set of a vector normed space into a vector normed space, derivative of a composed function, partial derivatives, Jacoby matrix, derivative of .
• Mean-value theorem and its applications, primitive of a regular function and derivation under the sum signe.
• Local inversion theorem and implicit functions theorem.
• Derivatives of superior order and Taylor formula.
• Local extremas , extrema of implicit functions and tied extrema.