INQ1097768  CALCULUS 2 20232024
Topic outline


Introduction to differential equations: Malthus equation.

Malthus equation: solution, Cauchy problem. Toward a more realistic demographic model: logistic equation.

Logistic equation: solution and application to population evolution. Catenary problem: setup and derivation of catenary equation.

First order linear equations. Homogeneous equations: general integral and examples. Non homogeneous equations: Lagrange trick and general integral. Example.

Exercises on first order linear equations. Cauchy problem, existence and uniqueness, examples. Separable variables equations: first steps.

Separable variables equations, examples, stationary solution and characterisation of solutions.

Exercises on separable variables equations. Second order linear equations: motivating examples, homogeneous equations, characteristic equation.

General solution of linear homogeneous second order equations. Fundamental system of solutions. Examples. Non homogeneous equations: Lagrange formula for particular solutions, general. Solution.

Exercises on second order linear equations. Cauchy problem: existence and uniqueness.
Euclidean space Rd: definition and vector space structure.

Euclidean norm. Definition of limit for a sequence of vectors. Examples. Component wise convergence. Properties of Euclidean norm: triangular inequality and Cauchy Schwartz inequality.

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)

Calculus of limits for functions of several variables: sections, non existence criterium, use of polar coordinates to show existence.

Exercises on limits for functions of several variables.

Exercises on limits at infinity.
Continuous functions, definition and main properties. Basic topology: open and closed balls.

Exercises on topology. Interior points, examples. Open sets, examples. Boundary points, examples. Closed sets, examples.

Cantor characterisation of closed sets. Sets defined by inequalities/equalities. Examples. Weierstrass theorem.

Exercises on topological properties. Weierstrass theorem on closed and unbounded domains.
Directional derivative: definition and examples.

Exercises on directional derivative. Partial derivatives, examples. Differentiability: definition, jacobian matrix. Gradient vector.

Exercises on differentiability. Remarks on differentiability: continuity, directional derivability. Differentiability test, examples. Chain rule, total derivative formula

Exercises on differentiability. Remarks on differentiability: continuity, directional derivability. Differentiability test, examples. Chain rule, total derivative formula

Fermat theorem and stationary points. Exercises on stationary points and optimization problems.

Exercises on optimization.

Constrained optimization: introduction, Lagrange multiplier theorem, example.

Exercises and remark on Lagrange multiplier theorem. Submersions. Extension to functions of several variables. Exercises. General Lagrange multipliers theorem.

Exercises on constrained optimization.
(Modified on dec 7)

Introduction to vector fields: examples. Potentials of a vector field, examples. Schwartz property and irrotational fields. Irrotational fields are not necessarily conservative: example.

Introduction to vector fields: examples. Potentials of a vector field, examples. Schwartz property and irrotational fields. Irrotational fields are not necessarily conservative: example.

Exercises on vector fields and potentials. Arc integral: exercises. Fundamental theorem of calculus for vector fields. Null circulations property and characterisation of conservative fields

Exercises on vector fields and potentials.
Introduction to multiple integration.

Definition of multiple integral and main properties. Reduction formula, examples.

Exercises on reduction formula. Integrability test through iterated integrals. Examples and exercises.

Exercises on reduction formula. Multiple integrals, reduction formula. Examples. Measure of a set. Exercises.

Change of variable for multiple integrals. Exercises. Integration in polar coordinates. Exercises.

Exercises on multiple integration in polar, spherical and cylindrical coordinates.

Exercises on multiple integration.
Functions of complex variable. Examples. Powers series, radius of convergence.

Exercises on power series. Exp, sh, ch, sin and cos. Complex logarithm. Exercises.

Exercises on power series. Exp, sh, ch, sin and cos. Complex logarithm. Exercises.

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.

CauchyRiemann equations. Exercises.




Update: Feb 11, 2024
This file contains assignments given during the AY 202223, and 202324.
NEW: solution of the Second Final Exam (February 2024).