Complements of Linear Algebra
Matrixes as representations of linear transformations.
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Eigenvalues and eigenvectors of a square matrix. Eigenspaces and algebraic multiplicity of an eigenvalue.
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Characteristic polynomial and geometric multiplicity of an eigenvalue.
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Eigenvalues of symmetric matrixes.
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Application to the classification of quadratic forms.
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Sequences and Elementary topology of
Euclidian distance; neighborhoods and open balls; interior, exterior, boundary; Open subsets of
R
n .
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Closure of a set and closed sets. Compact sets. Sequences in of : bounded sequences and convergent sequences.
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Topology from the point of view of sequences.
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Functions of several real variables
Some generalities: domain, range, graph and level curves.
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Limit of a real function of several variables. The algebra of limits.
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Som classical theorems.
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Limit with respect to a subset of the domain; Directional limits and main properties.
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Notion of continuity. Operations with continuous functions. The Weierstrass Theorem.
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Differential calculus
Directional derivatives and partial derivatives: definition and geometric interpretation.
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Linear approximation of a function: the notion of differentiability.
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Differentiability of vector valued functions. The Jacobian matrix.
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Partial derivation and function composition: the chain rule.
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Link between differentiability and the existence of directional and partial derivatives.
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Space of continuously differentiable functions. The Schwarz Theorem.
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Hessian Matrix and Taylor's formula.
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Application of the differential calculus to optimization
Critical points. Extremal values and saddle points.
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Second order criteria for the classification of critical points in an open set.
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Higher order criteria.
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Extrema with constraints and Lagrange multipliers.
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Study if extremal functions of a continuous function in a compact set.
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Integral calculus in
R
2
Definition and first properties.
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The Fubini theorem. Integration over simple regions. Application to the calculus of areas and volumes.
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Brief notions of Integration in
R
n
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Differential equations
First definitions and examples.
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First order differential equations.
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Linear second order differential equations.
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Non-homogeneous second order equations. The variation of constants method.
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