INQ1097768 - CALCULUS 2 2023-2024
Topic outline
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Introduction to differential equations: Malthus equation.
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Malthus equation: solution, Cauchy problem. Toward a more realistic demographic model: logistic equation.
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Logistic equation: solution and application to population evolution. Catenary problem: setup and derivation of catenary equation.
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First order linear equations. Homogeneous equations: general integral and examples. Non homogeneous equations: Lagrange trick and general integral. Example.
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Exercises on first order linear equations. Cauchy problem, existence and uniqueness, examples. Separable variables equations: first steps.
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Separable variables equations, examples, stationary solution and characterisation of solutions.
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Exercises on separable variables equations. Second order linear equations: motivating examples, homogeneous equations, characteristic equation.
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General solution of linear homogeneous second order equations. Fundamental system of solutions. Examples. Non homogeneous equations: Lagrange formula for particular solutions, general. Solution.
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Exercises on second order linear equations. Cauchy problem: existence and uniqueness.
Euclidean space Rd: definition and vector space structure.
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Euclidean norm. Definition of limit for a sequence of vectors. Examples. Component wise convergence. Properties of Euclidean norm: triangular inequality and Cauchy Schwartz inequality.
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Exercises on limits for sequences and norms. Infinite limits. Examples. Other norms: the Manhattan norm. Accumulation points. Limit of a function of vector variable vector valued. Example.
(This file has been integrated with the continuation of last example done on Oct 30)
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Calculus of limits for functions of several variables: sections, non existence criterium, use of polar coordinates to show existence.
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Exercises on limits for functions of several variables.
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Exercises on limits at infinity.
Continuous functions, definition and main properties. Basic topology: open and closed balls.
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Exercises on topology. Interior points, examples. Open sets, examples. Boundary points, examples. Closed sets, examples.
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Cantor characterisation of closed sets. Sets defined by inequalities/equalities. Examples. Weierstrass theorem.
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Exercises on topological properties. Weierstrass theorem on closed and unbounded domains.
Directional derivative: definition and examples.
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Exercises on directional derivative. Partial derivatives, examples. Differentiability: definition, jacobian matrix. Gradient vector.
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Exercises on differentiability. Remarks on differentiability: continuity, directional derivability. Differentiability test, examples. Chain rule, total derivative formula
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Exercises on differentiability. Remarks on differentiability: continuity, directional derivability. Differentiability test, examples. Chain rule, total derivative formula
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Fermat theorem and stationary points. Exercises on stationary points and optimization problems.
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Exercises on optimization.
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Constrained optimization: introduction, Lagrange multiplier theorem, example.
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Exercises and remark on Lagrange multiplier theorem. Submersions. Extension to functions of several variables. Exercises. General Lagrange multipliers theorem.
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Exercises on constrained optimization.
(Modified on dec 7)
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Introduction to vector fields: examples. Potentials of a vector field, examples. Schwartz property and irrotational fields. Irrotational fields are not necessarily conservative: example.
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Introduction to vector fields: examples. Potentials of a vector field, examples. Schwartz property and irrotational fields. Irrotational fields are not necessarily conservative: example.
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Exercises on vector fields and potentials. Arc integral: exercises. Fundamental theorem of calculus for vector fields. Null circulations property and characterisation of conservative fields
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Exercises on vector fields and potentials.
Introduction to multiple integration.
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Definition of multiple integral and main properties. Reduction formula, examples.
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Exercises on reduction formula. Integrability test through iterated integrals. Examples and exercises.
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Exercises on reduction formula. Multiple integrals, reduction formula. Examples. Measure of a set. Exercises.
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Change of variable for multiple integrals. Exercises. Integration in polar coordinates. Exercises.
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Exercises on multiple integration in polar, spherical and cylindrical coordinates.
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Exercises on multiple integration.
Functions of complex variable. Examples. Powers series, radius of convergence.
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Exercises on power series. Exp, sh, ch, sin and cos. Complex logarithm. Exercises.
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Exercises on power series. Exp, sh, ch, sin and cos. Complex logarithm. Exercises.
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Exercises on complex equations. Remarks on C logarithm. C differentisbility of power series. Derivatives of exp, sinh, cosh, sin and cos. Examples of functions which are not C differentiable. Cauchy Riemann equations.
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Cauchy-Riemann equations. Exercises.
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Update: Feb 11, 2024
This file contains assignments given during the AY 2022-23, and 2023-24.
NEW: solution of the Second Final Exam (February 2024).