Mon 08:15 - 10:00 in ETA F 5
Wed 08:15 - 10:00 in HG F 7 with Livestream in HG F 5
Thu 16:15 - 18:00 in ETA F 5
| Week # | Lecture # | Date | Topics | Notes |
|---|---|---|---|---|
| 1 | 1 | 19.02 | Rules and Webpage. Euclidean structure of \(\mathbb{R}^n\). Definition and Examples of Metric Spaces. Limits of sequences in Metric Spaces. | 9.1.1 |
| 2 | 21.02 | Subsequences in Metric Spaces. Convergence in \(\mathbb{R}^n\). Cauchy sequences and complete metric spaces. | 9.1.3 | |
| 3 | 22.02 | Open and closed sets, interior, boundary, closure in metric spaces. Characterization of open and closed sets with sequences. Continuity in metric spaces: three equivalent definitions. | 9.2.1, 9.2.2 | |
| 2 | 4 | 26.02 | Proof of the equivalence of the three continuity definitions. Lipschitz and uniformly continuous functions. Banach's fixed point theorem. Compactness: three equivalent definitions. | 9.2.2, 9.2.3, 9.2.4 |
| 5 | 28.02 | Proof of the equivalence of the three compactness definitions. | 9.2.4 | |
| 6 | 29.02 | Closedness VS Compactness. In \(\mathbb{R}^n\) a set is compact if and only if it is closed and bounded (Heine-Borel theorem). Continuous functions map compact sets to compact sets. Weierstrass theorem. Definition of connected set and path-connected set. A subset of \(\mathbb{R}\) is connected if and only if it is an interval. Continuous functions map connected sets to connected sets. | 9.2.4, 9.2.5, 9.2.6 | |
| 3 | 7 | 04.03 | Path connected sets are connected. Equivalence between connectedness and path-connectedness for open subsets of \(\mathbb{R}^n\). Continuous functions are uniformly continuous on compact sets (Heine-Cantor Theorem). Normed vector spaces. Example: the \(p\) norm in \(\mathbb{R}^n\). Norms induced by scalar products. Cauchy-Schwarz inequality. Definition of equivalent norms. Theorem: all norms in \(\mathbb{R}^n\) are equivalent (only statement). | 9.2.6, 9.3.1, 9.3.2, 9.3.3 |
| 8 | 06.03 | Functions of several variables. Heuristics: four interpretations of the derivative of a function of one variable. Definition of differential and directional derivative. Differentiable implies existence and linearity of directional derivatives. Existence and continuity of all directional derivatives implies differentiability. | 10.1.1, 10.1.2 | |
| 9 | 07.03 | Notation \(C^1(U),C^1(U,\mathbb{R}^m)\). Jacobi matrix. Interlude: the Hilbert-Schmidt norm of a matrix. Theorem: the chain rule. The mean value theorem. Locally, differentiable functions are Lipschitz continuous. Functions with vanishing differential. | 10.1.2, 10.1.3, 10.1.4 | |
| 4 | 10 | 11.03 | Higher order derivatives. Notation \(C^k(U),C^k(U,\mathbb{R}^m)\). Schwarz's Theorem. Multi-indeces notation. Taylor's forumla in several variables. | 10.2.1, 10.2.2 |
| 11 | 13.03 | Proof of the Taylor's formula from the one from Analysis I. Practical computation of Taylor expansions. Real-analytic functions in several variables: estimate on the derivative and proof of the convergence of the series. | 10.2.3, 10.2.4 | |
| 12 | 14.03 | Unique continuation for analytic functions. Gradient of a function. The gradient vanishes at local extrema. Constrained minimisation, Lagrange multipliers. | 11.1.1, 11.1.2 | |
| 5 | 13 | 18.03 | The spectral Theorem for symmetric matrices, proof with Lagrange multipliers. The Hessian matrix and the Hessian test at a critical point. | 11.2, 11.3 |
| 14 | 20.03 | Variational proof of the fundamental Theorem of Algebra. Convex sets and convex functions. Convexity for \(C^2\) and \(C^1\) functions. General Jensen inequality. | 11.4, 11.5 | |
| 15 | 21.03 | Lipschitz perturbation of the identity and other preliminary Lemmas for the inverse function theorem. | 12.1 | |
| 6 | 16 | 25.03 | The inverse function theorem. Diffeomorphisms beween open sets in \(\mathbb{R}^n\). Implicit function Theorem and definition of submanifolds of \(\mathbb{R}^n\). | 12.1 |
| 17 | 27.03 | Three equivalent ways to give a submanifold: parametric, cartesian, graphical. Example of the sphere and the torus. | 12.2 | |
| 18 | 28.03 | Dyadic cubes and itervals and their refinements. The measure of dyadic sets. Inner and outer measure of general sets and Jordan-measurable sets. Null sets in the sense of Jordan and Lebesgue, equivalence of the two notions for compact sets. | 13.1 | |
| Holidays | Holiday | 01.04 | - | - |
| Holiday | 03.04 | - | - | |
| Holiday | 04.04 | - | - | |
| 7 | 19 | 08.04 | Bounded sets ar eJordan measurable if and only if their boundary is Lebesgue null. ``Sandwich'' criterion for Jordan measurability. Lipschitz maps preserve null sets. Graphs of (uniformly) continuous functions are Jordan measurable. Theorem: the Jordan measure is additive, it assigns 1 to the unit cube and has the expected value on rectangles with sides parallel to the coordinate axis. | 13.1 |
| 20 | 10.04 | The unit ball is Jordan measurable. Theorem: the Jordan measure is invariant under rotation. Proof via polar decomposition of linear maps and homogeneity/scaling considerations. | 13.1 | |
| 21 | 11.04 | Lecture cancelled | - | |
| 8 | 22 | 15.04 | Riemann-integrable-functions, linearity of the integral, positive and negative parts, integral as area of the hypograph. Proposition: Uniformly continuous functions are Riemann-integrable. Theorem: Change of Variables formula in multiple integrals (with proof). | 13.2 |
| 23 | 17.04 | Slicing formula for Jordan measurable sets. Fubini's Theorem for continuous functions. | 13.3 | |
| 24 | 18.04 | Theorem: Differentiation under the integral sign. Examples of computations: change of variables and Fubini. Spherical coordinates in 3D. Definition of improper integrals for non-negative functions, well-posedness of the definition. | 13.3, 13.4,13.5 | |
| 9 | 25 | 22.04 | Length of a curve. Lemma: length as total variation (no proof). Definition: Isometries of the Euclidean and Gram determinants. Definition: m-Volume of a parametrized submanifold. Heuristic motivation. Lemma: this definition is well-posed (i.e., invariant by reparametrisations). Example: the formula for the 2-volume of surfaces in 3D (relationship with the vector product). | 13.6 |
| 26 | 24.04 | Example: the 2-volume of the round sphere. Integration of functions on manifolds: the case of functions supported in one parametrisation. Lemma: the definition is well-posed in this case. Lemma: the m-volume of a manifold when parametrised graphically. Lemma: \(C^\infty\) partitions of unity. | 13.6, 13.7 | |
| 27 | 25.04 | Definition: graphical covers of a manifold and partitions of unity subordinated to such a cover. Integration of functions on manifolds: the general case. (not the proof of well-posedness). Definition of tangent and normal vectors to a manifold. Definition of bounded domain with \(C^k\) boundary. | 13.7, 14.4.1 | |
| 10 | 28 | 29.04 | Proposition: formula of the unit normal to a graph. Lemma: Fubini Theorem for graphical domains (no proof). Integration by parts formula: the case of local graphs. | 14.1.1, 14.1.2 |
| Holiday | 01.05 | - | - | |
| 29 | 02.05 | Integration by parts formula: the general case. Proof using the local version for graphs and partitions of unity. Definition: divergence of a vector field and the the Divergence Theorem and its equivalence to the integration by parts formula. | 14.1.3 | |
| 11 | 30 | 06.05 | Definitions: piecewise \(C^1\) paths and work of a vector field along them. Lemma: the work does not depend on the parametrisation. Definition: potential for a vector field in some domain. Work as difference of potential. Lemma: necessary integrability conditions to admit a potential. Defintion: Homotopy of paths and simply connected domains. Theorem: Poincare' Lemma. | 14.2.1, 14.2.2 |
| 31 | 08.05 | Proof of Poincare' Lemma in convex domains and proof of the general case. Stokes theorem in 3D: definition of oriented surface and statement of the theorem (extra material). Stokes Theorem for immersed disks. Interlude: skew-symmetric 3x3 matrices, infinitesimal rotations and the cross product. | 14.2.2, 14.3 | |
| Holiday | 09.05 | - | - | |
| 12 | 32 | 13.05 | Proof of Stokes Theorem for an immersed disk. Defintion of a m-form on \(\mathbb{R}^n\) as a natural parametrisation-invariant object to integrate over m-surfaces. Fundamental properties of differential forms: behaviour under change of variables (pull-back), existence of the exterior differential (that generalizes div, curl etc). | 14.3, 14.4 |
| 33 | 15.05 | Linear ODEs of arbitrary order with constant coefficients. Initial value problem and affine structure of the set of solutions. Existence of a solution to the homogeneous initial value problem using poly-exponentials and the characteristic polynomial of the equation. Gronwall's Lemma. There are \(m\) linear independent solutions. Uniqueness of solutions using Gronwall's Lemma. | 15.1 | |
| 34 | 16.05 | The in-homogeneous problem \(L y =f\), the case of \(f(t)=q(t)\exp(\alpha t)\). Abstract procedure to find the solution. Example: the harmonic oscillator forced by \(f(t) = \sin(\omega t)\). Resonance at critical frequency. The case of a general \(f\): Duhamel's representation formula in terms of the "Pulse at zero" solution. Exercise on integrals derivatives of parametric integrals. | 15.1 | |
| 13 | Holiday | 20.05 | ||
| 35 | 22.05 | |||
| 36 | 23.05 | |||
| 14 | 37 | 27.05 | ||
| 38 | 29.05 | |||
| 39 | 30.05 |
Mon 10:15 - 12:00 in CAB G 56
Wed 16:15 - 17:00 in LEE C 114
Mon 10:15 - 12:00 in CHN D 42
Thu 15:15 - 16:00 in NO D 11
Mon 10:15 - 12:00 in CHN D 46
Fri 13:15 - 14:00 in ML J 34.1
Mon 10:15 - 12:00 in ETZ E 8
Thu 15:15 - 16:00 in LEE D 105
Mon 10:15 - 12:00 in ETZ E 9
Tue 13:15 - 14:00 in CHN D 48
Mon 10:15 - 12:00 in HG E 33.3
Fri 12:15 - 13:00 in HG G 26.5
Mon 10:15 - 12:00 in HG E 33.5
Thu 15:15 - 16:00 in CAB G 52
Mon 10:15 - 12:00 in HG G 26.3
Thu 15:15 - 16:00 in CAB G 56
Mon 10:15 - 12:00 in ML J 37.1
Thu 15:15 - 16:00 in CAB G 59
Mon 10:15 - 12:00 in LFW E 13
Thu 15:15 - 16:00 in ML F 38
Mon 10:15 - 12:00 in ML H 43
Thu 15:15 - 16:00 in HG G 26.3
Mon 10:15 - 12:00 in ML J 34.3
Thu 15:15 - 16:00 in LFW C 11
Mon 12:15 - 14:00 in CHN F 46
Thu 15:15 - 16:00 in LFW C 4
Mon 16:15 - 18:00 in CAB G 59
Fri 12:15 - 13:00 in CLA E 4
Mon 16:15 - 18:00 in LEE C 104
Tue 13:15 - 14:00 in HG G 26.5
Mon 16:15 - 18:00 in CAB G 52
Fri 12:15 - 13:00 in CAB G 56
Mon 16:15 - 18:00 in CHN E 42
Fri 12:15 - 13:00 in ML J 34.1
Mon 16:15 - 18:00 in LFW C 11
Fri 13:15 - 14:00 in HG G 26.5
Mon 16:15 - 18:00 in NO D 11
Fri 13:15 - 14:00 in ETZ E 7
Mon 16:15 - 18:00 in ML J 37.1
Wed 16:15 - 17:00 in HG E 33.3
Mon 16:15 - 18:00 in LEE C 114
Fri 13:15 - 14:00 in ML F 38