MATH3011 and MATH4201

**Lecturer :** Norbert Peyerimhoff

**Term :** Michaelmas 2011/12 and Epiphany 2012

- Mondays, 9:00 in PH8
- Wednesdays, 9:00 in PH8

**Problem Classes :**

- Thursday, 20 October 2011, 5pm in CG85
- Thursday, 17 November 2011, 5pm in CG85
- Thursday, 9 February 2012, 5pm in CG85
- Thursday, 8 March 2012, 5pm in CG85

**Literature**

Books relevant for the first term:

- E.T. Copson,
*Metric Spaces*. Cambridge University Press. - J. Dieudonne,
*Foundations of modern analysis*. Academic Press. - E. Hewitt, K. Stromberg,
*Real and abstract analysis*. Springer Verlag. - K. Hoffman,
*Analysis in Euclidean space*. Prentice-Hall. - M.P. do Carmo,
*Differential Forms and Applications*. Springer Verlag.

- M.P. do Carmo,
*Differential Forms and Applications*. Springer Verlag. - J.R. Munkres,
*Analysis on manifolds*. Westview Press. - M. Spivak,
*Calculus on manifolds*. Benjamin.

A roadmap on the 4H extra reading material is here. The references for the extra reading material are

- The "Notes on the Arzela-Ascoli Theorem" on Joel Shapiro's Page
- The two Sections 6.4 Completeness and 6.5 Compactness, i.e., pages 263-285 in
K. Hoffman's book
*Analysis in Euclidean space*

**Assignments**

Homework | Date | Hand in | Solutions |

Exercise Sheet 1 pdf | 10.10.2011 | --- | Solution Sheet 1 pdf |

Exercise Sheet 2 pdf | 17.10.2011 | --- | Solution Sheet 2 pdf |

Exercise Sheet 3 pdf | 24.10.2011 | --- | Solution Sheet 3 pdf |

Exercise Sheet 4 pdf | 31.10.2011 | --- | Solution Sheet 4 pdf |

Exercise Sheet 5 pdf | 7.11.2011 | 16.11.2011 | Solution Sheet 5 pdf |

Exercise Sheet 6 pdf | 16.11.2011 | --- | Solution Sheet 6 pdf |

Exercise Sheet 7 pdf | 23.11.2011 | --- | Solution Sheet 7 pdf |

Exercise Sheet 8 pdf | 30.11.2011 | --- | Solution Sheet 8 pdf |

Exercise Sheet 9 pdf | 7.12.2011 | 14.12.2011 | Solution Sheet 9 pdf |

Exercise Sheet 10 pdf | 14.12.2011 | --- | Solution Sheet 10 pdf |

Exercise Sheet 11 pdf | 18.1.2012 | --- | Solution Sheet 11 pdf |

Exercise Sheet 12 pdf | 25.1.2012 | --- | Solution Sheet 12 pdf |

Exercise Sheet 13 pdf | 1.2.2012 | 8.2.2012 | Solution Sheet 13 pdf |

Exercise Sheet 14 pdf | 8.2.2012 | --- | Solution Sheet 14 pdf |

Exercise Sheet 15 pdf | 22.2.2012 | 5.3.2012 | Solution Sheet 15 pdf |

Exercise Sheet 16 pdf | 29.2.2012 | --- | Solution Sheet 16 pdf |

**Content of Lectures**

Date | Content |

Monday, 10 October 2011 (Week 1) | Definition of a metric space, examples, recall supremum definition, Cauchy sequences and convergent sequences, meaning of completeness of a metric space, how to get from the rationals to the real numbers |

Wednesday, 12 October 2011 (Week 1) | Inner product spaces, examples, Cauchy-Schwarz inequality with proof, how to prove triangle inequality with Cauchy-Schwarz, convergence implies Cauchy, space of bounded functions with supremum distance is complete |

Monday, 17 October 2011 (Week 2) | Example of vector space of bounded functions and its completeness w.r.t. supremum distance, open and closed balls in metric spaces, openness and closedness of sets, boundary of a set, properties of open and closed sets |

Wednesday, 19 October 2011 (Week 2) | Equivalent formulations for closed sets, closed subset of a complete space is again complete, sequential compactness, open covers and compactness, closed subsets of compact sets are compact |

Monday, 24 October 2011 (Week 3) | Heine-Borel Theorem, definition of continuity and uniform continuity, continuity via convergent sequences, properties of continuity (pre-images of open/closed sets are closed, images of compact sets are compact), definition of a normed vector space and of a Banach space, example of continuous functions and bounded continuous functions, space of continuous functions on a compact set is a Banach space |

Wednesday, 26 October 2011 (Week 3) | Examples of norms in R^n, all norms of R^n are equivalent, space of continuous functions with L2-integral norm is not a Banach space, definition of a Hilbert space, example of L2[a,b], linear and bounded linear operators, norm of bounded linear operators, matrices as bounded linear operators in finite-dimensional vector spaces, examples of vector spaces of bounded linear operators, properties of operator norm |

Monday, 31 October 2011 (Week 4) | B(V,W) Banach space if W Banach space, definition of a contraction, Banach's Contraction Mapping Principle with proof, definition of totally differentiable and matrix of partial derivatives, examples: gradient and tangent vectors of curves, totally differentiable functions are continuous, function with continuous partial derivatives is totally differentiable, example of a non-continuous function with well defined partial derivatives, Chain Rule |

Wednesday, 2 November 2011 (Week 4) | Higher derivatives, definition of C^k, Mean Value Theorem with proof and with consequences, definition of a diffeomorphism, derivative of inverse function, Cramer's Rule |

Monday, 7 November 2011 (Week 5) | Useful facts about the space of invertible matrices (like inv: GL(n,R) -> GL(n,R), A -> A^(-1) is a diffeomorphism, if norm(A) < 1 then Id - A invertible), Inverse Function Theorem with proof |

Wednesday, 9 November 2011 (Week 5) | Proof of the useful facts about the space of invertible matrices given in the previous lecture, geometric motivation for the Implicit Function Theorem |

Monday, 14 November 2011 (Week 6) | Algebraic motivation for the Implicit Function Theorem, Statement of Implicit Function Theorem, Explicit formula for the derivative of the implicit function, length of a curve, piecewise C^k curves and calculation with help of tangent vectors |

Wednesday, 16 November 2011 (Week 6) | Proof of length calculation with help of tangent vectors, example of helix, gradient as vector field, vector fields and integral curves, example of rotational vector field, integral curves as solutions of vector valued homogeneous differential equations, relation between gradient and level sets |

Monday, 21 November 2011 (Week 7) | Differentials as special differential 1-forms, vector space Omega^1(U) of differential 1-forms, differential 1-forms and defining vector field, dx_j as basis for differential 1-forms, example for calculations with differential 1-forms, exact 1-forms, line integral of a 1-form, example of a line integral |

Wednesday, 23 November 2011 (Week 7) | Invariance of line integral under orientation preserving reparametrisations, line integral of a differential, geometric critera for exactness of 1-forms and (via line integrals) with proof, example how to apply this to conclude that a form is not exact |

Monday, 28 November 2011 (Week 8) | Preparation to introduce differential k-forms, properties of the determinant (multilinear and alternating), definition of alternating k-forms, vector space Lambda^k(V) of alternating k-forms, differential k-forms, vector space Omega^k(U) of differential k-forms, wedge-profuct, basis and dimension of vector space of alternating l-forms, definition of Lambda^0(V) and Lambda^k(V) for k greater than dimension of V |

Wednesday, 30 November 2011 (Week 8) | Proof of statement about basis and dimension of vector space of alternating l-forms, extension of wedge-product to k-forms and associativity of this multiplication, wedge-product on differential forms, example of evaluation of differential 2-form and of wedge-product of differential forms |

Monday, 5 December 2011 (Week 9) | Effect of commuting differential in a wedge product (a factor (-1)^kl shows up), definition of the exterior differential of a differential form, example of taking exterior differential, differential of a 1-form and derivation that closedness (introduced in an exercise) agrees with the 1-form having vanishing differential, properties of d (linearity, a non-commutative product rule and dd=0) |

Wednesday, 7 December 2011 (Week 9) | Definition of closed and exact differential k-forms, definition of starlike, Poincare Lemma, homotopy, integration of closed forms along homotopic curves agrees, free homotopy, integration of closed forms along closed freely homotopic curves agrees |

Monday, 12 December 2011 (Week 10) | Proof that integration of closed forms along homotopic curves agrees, definition of the pullback of a differential form, properties of the pullback (linearity, pullback of a differential, pullback of a wedge-product equals wedge-product of pullback |

Wednesday, 14 December 2011 (Week 10) | Example of a pullback, composition of pullbacks, pullback commutes with exterior differential, definition of k-th deRham cohomology, example of deRham cohomology of R^2-0 |

Monday, 16 January 2012 (Week 11) | Integration in R^n, sets of measure zero, example, characterisation of Riemann integrable functions, examples |

Wednesday, 18 January 2012 (Week 11) | Fubini's Theorem, Transformation Rule, integral of differential forms, invariance of integral of differential forms under pullbacks, volume form, example |

Monday, 23 January 2012 (Week 12) | Theorem of Picard Lindeloef with iterations, proof of the theorem by reducing it to contraction mapping principle, Example |

Wednesday, 25 January 2012 (Week 12) | Definition of a k-dimensional submanifold in R^n, examples, coordinate changes, example of parametrisation of S^2 by two coordinate patches (stereographic projection) |

Monday, 30 January 2012 (Week 13) | Continuation fo stereographic projection example, manifolds are locally graphs, illustration, singular/regular points and regular values manifolds as preimages of regular values |

Wednesday, 1 February 2012 (Week 13) | Example of a regular value, preimages of regular values are manifolds, examples of manifolds as preimages of regular values (e.g., one sheeted and two sheeted hyperboloid), almost global coordinate patches, example, tangent vector and tangent space of a manifold |

Monday, 6 February 2012 (Week 14) | explicit calculation of tangent spaces if manifold is given by coordinate patch or implicitly by equation, example, critical points of functions on manifolds, local minima and maxima on manifolds, method of Langrangian multipliers with proof |

Wednesday, 8 February 2012 (Week 14) | Continuation of proof of method of Lagrangian multipliers, example, distance between closed and compact sets |

Monday, 13 February 2012 (Week 15) | Another example with Lagrangian multipliers, definition of an orientable manifold, example, definition of an oriented basis in a vector space, standard orientation in R^n, oriented manifolds and oriented bases in tangent spaces, hypersurfaces and unit normal vectors, notion of positive orientation of a unit normal vector |

Wednesday, 15 February 2012 (Week 15) | Global normal vector of hypersurface in case of preimage of a regular value, Moebius Strip - example of a nonorientable manifold, differential forms on manifolds, global differential form on manifold as compatible family of differential forms on coordinate patches via pullback |

Monday, 20 February 2012 (Week 16) | Definition of exterior differential for differential forms on manifolds, integration of a differential form on a manifold (in case that differential form only nonzero on one coordinate patch), independence of definition of coordinate patch, partition of unity, integration of a general differential form on a manifold using a partition of unity, independence of this definition of choice of partition of unity |

Wednesday, 22 February 2012 (Week 16) | Integration example with an almost global coordinate patch, preparations to introduce manifolds with boundary, interior and boundary points, boundary of a k-dimensional manifold as a (k-1)-dimensional manifold without boundary |

Monday, 27 February 2012 (Week 17) | Coordinate patches for the boundary, induced orientation of the boundary, modification of the tangent space in case of manifolds with boundary, examples |

Wednesday, 29 February 2012 (Week 17) | Stokes Theorem, example, proof of Stokes Theorem |

Monday, 5 March 2012 (Week 18) | Continuation of proof of Stokes Theorem, Corollary about integral of an exact form over a closed manifold, reduction of Stokes for differential forms to Classical Theorem of Stokes |

Wednesday, 7 March 2012 (Week 18) | Reminder of definition of deRham cohomology and Betti numbers, cohomological implication of Poincare's Theorem and interpretation of zeroeth deRham cohomology, definition of an exact sequence, special cases of exact sequences, vanishing of the alternating sum of vector space dimensions in exact sequences with proof by induction, short Mayer-Vietoris sequence |

Modday, 12 March 2012 (Week 19) | Long exact Mayer-Vietoris sequence, example: deRham cohomology of R^2-0, definition of homotopic maps, homotopy equivalence, contractibility, examples, invariance of cohomology for homotopy equivalent sets |

Wednesday, 14 March 2012 (Week 19) | Example: DeRham cohomology of S^n minus n distinct points, Poincare Duality, classification of oriented closed surfaces of genus g and their deRham cohomology, Euler characteristic, connections between Analysis and Differential Geometry, Complex Analysis, AMV, Algebraic Topology and Algebraic Geometry. |

Last modified: 15.3.2012