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Frechet differentiation Description

Key Elements

Code

MATH 302

Formation

BS Mathematics

Semester

5

Credits

6

Number of Teaching Hours

30

Number of Tutoring Sessions

30

Number of Laboratory Sessions

0

Content

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.