CALCULUS 2 2025-2026 - INQ1097768
Section outline
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Exercises on norms and limits of sequences. Infinite limit. Definition of limit for a vector valued function of vector variable.
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Test for non existence of limit of a function. Examples and exercises.
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Exercises on calculus of limits using polar and spherical coordinates.
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Exercises on calculus of limits. Limits at \(\infty_d\) and infinite limits.
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Exercises on calculus of limits. Definition of continuous function. Examples.
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Basic tolopogical definitions: open and closed ball, interior and boundary points, open and closed sets. Examples and exercises. Cantor's characterization of closed sets.
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Topological properties of sets defined by inequalities and equalities. Bounded and unbounded sets. Examples. Compact sets. Weierstrass' theorem. Exercises.
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Exercises on open, closed, bounded, compact sets of R^d. Existence of min/max for continuous functions on closed and unbounded domains of R^d. Definition of directional derivative. Examples and exercises.
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Partial derivatives: definition and examples. Differentiability, jacobian matrix and connection with directional differentiability.
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Remarks on differentiability: directional differentiability and continuity. Gradient vector. Exercises: checking differentiability. Test of differentiability.
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Exercises on differentiability. Local min/max points. Stationary points. Examples.
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Exercises on stationary points. Fermat's theorem: remarks and proof. Optimization problems: exercises.
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Convexity and concavity, sufficiency of the first order condition. Second derivative: Hessian matrix, examples, Schwarz theorem. Positive/negative definite matrices.
(second part of the class missing because of a technical failure).
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Taylor's formula and classification of stationary points. Exercises.
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Constrained optimization. Lagrange multiplier theorem. Examples and exercises.
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Exercises on Lagrange's multiplier theorem. Submersion maps.
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Exercises on submersion maps and multiple constraints optimization.
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Vector fields: definition and examples. Conservative fields and potentials. Examples. Irrotational fields, examples and exercises.
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Exercises on vector fields and potentials. Line integral.
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Circulations and characterization of conservative fields through null circulations. Examples and exercises.
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Introduction to Multiple Integrals: motivations, and definition of the integral for positive and bounded functions.
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Definition of integral for a function of variable sign. Reduction formula: Fubini's theorem, examples.
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Exercises on reduction formula. Test of integrability: Tonelli's theorem. Examples and exercises.
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Exercises on reduction formula for double integrals. Reduction formula for triple integrals. Examples.
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Exercises reduction formula. Definition of area and volume, slicing formula. Volume of the sphere. Change of variable: recap from one variable calculus, change of variable formula for multiple integrals. Integration in polar coordinates.
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Exercises: computing integrals in polar coordinates. Spherical and cylindrical coordinates, exercises.
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Exercises on integration in spherical/cylindrical coordinates and general change of variables.
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Exercises on change of variable. Green's formula and irrotational fields.
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Exercises on Green's formula, area formula. Introduction to functions of complex variable.
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Series of complex numbers: definition of convergence, Geometric series, test for convergence, absolute convergence. Power series, radius of convergence, disk of convergence. Exponential function: definition and main properties.
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Exercises on power series. Elementary functions: sin, cos, sinh, cosh and their properties. \(\Bbb C-\)logarithm, existence and examples. Solving equations with elementary functions: exercises.
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Exercises on logarithms, equations with elementary functions. \(\Bbb C-\)differentiability: definition and first examples. Rules of calculus, derivative of a power series.
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Differential Equations: definition of solution, Cauchy problem. Existence and uniqueness: examples, example of non uniqueness.
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Local existence and uniqueness. Qualitative study of solutions of differential equations. Exercises.
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Exercises on qualitative study of differential equations.
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