Lectures in course MATH-105(a): Advanced analysis II - vector analysis

Generalized Integrals: Definition and Applications

Covers the definition and applications of generalized integrals in advanced analysis, including real functions, differential equations, and multiple integrals.

Implicit Functions Theorem

Explains the Implicit Functions Theorem and conditions for defining functions implicitly.

Limits and Convergence

Covers the concept of limits and convergence, including the definition of limits and absolute convergence.

Advanced analysis II: implicit functions theorem

Covers the implicit functions theorem, vectorial functions, hypersurfaces intersection, and extrema of real functions.

Vector Spaces: Operations and Properties

Explores vector spaces, scalar products, standards, and symmetric matrices in the context of operations and properties.

Advanced analysis II: jordan-measurable sets

Explores Jordan-measurable sets and their properties, including volume calculations and change of variables in integrals.

Sequences and Series

Covers sequences, series, subsequence, convergence, open and closed sets in mathematics.

Existence and Uniqueness of Local Solutions

Explores the proof of existence and uniqueness of local solutions for a Cauchy problem with separable variables.

Sets and Closure

Covers open and closed sets, adhesion, and convergence in sets.

Integral-equations

Covers integral equations, including homogeneous and non-homogeneous equations, general solutions, and examples.

Limit Definition and Continuity

Covers the limit definition of a function and continuity on a set.

Advanced Analysis II: Homogeneous ODEs and Banach Spaces

Explores the resolution of ODEs and the Banach fixed-point theorem.

Advanced-analysis-ii

Explores advanced analysis topics, including Cauchy sequences, Banach spaces, and the Cauchy-Lipschitz theorem.

Differential Calculation: Notion of Derivative

Explores differential calculation and the notion of derivative in advanced analysis II.

Partial Derivatives and Applications

Explores partial derivatives, Jacobian matrix, directional derivatives, and their applications.

Cauchy-Lipschitz Theorem: Local Existence and Uniqueness

Explores the Cauchy-Lipschitz theorem for local existence and uniqueness of solutions to differential equations.

Local-existence-unicity-theorem

Covers the local existence and uniqueness theorem, focusing on the Cauchy problem and global solution conditions.

Derivability and Linear Applications

Explores derivability conditions, growth directions, and linear applications using matrices.

Advanced Analysis II: Integrals on Continuous Functions

Explores integrals on continuous functions and their properties, including uniform continuity.

Partial Derivatives: Understanding and Applications

Explores the computation and significance of partial derivatives in determining rates of change.