Fundamentals of mathematical analysis II
Topic outline
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This video introduces to the Course of Fundamentals of Mathematical Analysis 2.
Dates of classes correspond to AY 2021/22, see communications for AY 2022/23.
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Contains the Notes (Theory and Exercises) of the Entire Course
Course Syllabus
- Euclidean space R^d (Chapter 1)
- Differential Calculus (Chapter 2)
- Vector Fields (Chapter 3)
-Integration (Chapter 4)
- Surface Integration (Chapter 5)
- Basic Differential Equations (Chapter 6)
- Advanced Differential Equations (Chapter 7)
- Systems of Differential Equations (Chapter 8)
Detailed program will be given at the end of the course.
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Contains exercises to prepare for the exam.
Text of exercises 1-8, Solutions of exercises 1-8.
Text of exercises 9-10 (first partial exam) and solutions.
Text of exercises 11-22 (second partial exam).
Solutions to exercises 11-19.
(updated JUNE 6TH)
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Introduction to the course. Euclidean norm, definition and main properties.
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Limit of a sequence of vectors: finite and infinite limits, examples and exercises. Accumulation points, limit of a function of vector variable vector valued.
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Exercises on limits for sequences. Limits of functions: sections, non existence criterium. Exercises. Existence of limits.
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Exercises on limits for functions of several variables.
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Exercises on limits for functions of several variables.
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Exercises on limits.
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Continuity. Basic topological concepts: interior and boundary points. Open and closed sets. Examples.
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Exercises on open, closed, bounded and compact sets. Weierstrass theorem. Connected sets, intermediate values property. Directional derivative. Examples,
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Cantor characterisation of closed sets. Sets defined by constraints. Examples. Bounded sets.
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Remarks on directional derivative. Partial derivatives, examples. Differentiability, jacobian matrix, gradient vector, examples.
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Exercises on differentiability. Rules of calculus, total derivative. Fermat theorem, stationary points-
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Remarks on Fermat theorem and stationary points. Applications to optimization problems: exercises.
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Exercises on unconstrained optimization. Constrained optimization: Lagrange multiplier theorem for functions of 2 variables. Examples.
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Exercises on Lagrange multiplier theorem.
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General Lagrange multipliers theorem. Exercises
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Multiple integrals (slide interrupted by electricity shut down)
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Exercises on reduction formula. Change of variables formula. Integration in polar coordinates. Examples.
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Exercises on multiple integration with polar, spherical and cylindrical coordinates.
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Parametric surfaces, examples. Area of a parametric surface: examples and exercises.
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Exercises on areas of surfaces. Surface integrals. Flux of a vector field, examples. Outward flux: divergence theorem and applications. Exercises
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Exercises on divergence theorem and fluxes.
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Introduction to differential equations: Malthus and logistic equations, catenary equation, Newton equations.
First order linear equations: general integral, Cauchy problem, exercises.
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Exercises on first order linear equations.
Separable variables equations. Examples. Solution of logistic and catenary problems. Cauchy problem: exercises.
Second order linear equations with constant coefficients: general solution, Lagrange formula. Exercises.
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