Theory Analyse in meer variabelen (WISB 243)

See also exercises
Back to the home page.
Analyse in meer variabelen, period 4

All references are to the books by Duistermaat and Kolk

1a. Week 17 (April 24): 2.1 Linear algebra
  • linear algebra, distinction of linear map and matrix, Euclidean norm, norm as trace.
  • Lemma 2.1.1: multiplicative properties of this norm.
  • Inversion of matrices, Cramer s rule
    2.2 Differentiable mappings
  • total derivative, relation with directional derivative, Lemma 2.2.3.
  • Prop 2.2.9: differentiability component wise
    2.3 Directional and partial derivatives
  • Def 2.3.1, Prop. 2.3.2. criterion for differentiability: Thm 2.3.4, Def 2.3.6, Thm 2.3.7
    2.4 Chain rule
  • Lemma 2.27: Hadamard s characterization of differentiability, Cor. 2.2.9,
  • Theorem 2.4.1: chain rule, and proof by using Hadamard.
    • 1b. Week 17 (April 26):
  • Cor 2.4.2: chain rule for partials, Lemma 2.4.7. 2.5 Mean value theorem
  • Lemma 2.5.1 and Thm. 2.5.3: Lipschitz continuity
    3 Inverse and implicit function theorems
    3.1 Diffeomorphisms
    3.2 Inverse function theorem
  • the notion of diffeomorphism
  • inverse function theorem for homeomorphisms: Lemma 3.2.1 and Prop. 3.2.2, formulation.
    • 2a. Week 18 (May 1):
  • inverse function theorem for homeomorphisms: Lemma 3.2.1 and Prop. 3.2.2, formulation.
  • review of contraction theorem, Lemma 1.7.2.
  • Prop 3.2.3: local inverse is homeomorphism
  • the notion of local diffeomorphism, local inverse function theorem Theorem 3.2.4.
  • global inverse function theorem, 3.2.8.
  • 3.3: Application to polar coordinates.
    • 2b. Week 18 (May 3):
  • 3.4: description of implicit function question by parameter dependent system of equations.
  • idea of the linearized system
  • idea of implicit differentation
  • Thm 3.5.1: formulation and proof of implicit function theorem.
  • 3.6.A: Application: smooth dependence of roots of a polynomial.
  • application: hypersurface as local graph of a function
    • 3a. Week 19 (May 8):
  • Def 4.2.1 of submanifold of $\R^n$
  • Preferred definition, see extra notes.
  • Definition of immersion
  • immersion theorem, Thm 4.3.1, see also formulation in extra notes. Representation of submanifold by an immersion.
  • submersion theorem, Thm 4.5.2, see also formulation in lecture notes.
  • Characterization of submanifold by submersion, local and global.
    • 3b. Week 19 (May 10): no lecture, Hemelvaartsdag
    4a. Week 20 (May 15):
  • Proof of the immersion theorem.
  • Definition 4.2.4 of embedding; regular point; parametrization; local coordinatization; chart.
  • Corollary 4.3.2: image of embedding is submanifold.
  • Characterizations of Theorem 4.7.1 revisited.
  • The fibers of a global submersion are submanifolds (see extra notes).
  • smoothness lemma and smoothness of reparametrisations: Lemma 4.3.3 (see also extra notes).
    • 4b. Week 20 (May 17):
  • Ck functions on a Ck manifold, 4.7.3.
  • Ck maps between Ck manifolds.
  • 5.1: the tangent space, Definition, linearity,
  • characterizations Thm 5.1.2; skip Prop 5.1.3 and Cor. 5.1.4.
  • 5.2: tangent mapping, Def 5.2.1. Problem of generalization.
    • 5a. Week 21 (May 22):
  • 5.4: Application of tangent space: Lagrange multipliers.
  • start of chapter 6: integration: 6.1 - 6.3
  • 6.1: rectangles
  • 6.2 Riemann integration
  • 6.3 Jordan measurability
    • 5b. Week 21 (May 24): lecture by Fabian Ziltener
  • Jordan measure continued: Thm. 6.3.5
  • Corollary 6.3.8: graphs hence submanifolds are locally negligable
  • Def. 6.4.1: notion of step function
  • changing the order of integration; Thm 6.4.2.
  • Theorem 6.4.5 on successive integration.
    • 6a. Week 22 (May 29): lecture by Fabian Ziltener
  • Sketch of proof of Thm. 6.4.2: successive integration
  • Thm 6.6.1: change of variables theorem
  • Cor. 6.6.2: tranformation formula for Jordan volume
  • Example 6.6.4: polar coordinates
  • Example 6.10.8: integral of Gauss function
    • 6b. Week 22 (May 31): lecture by Fabian Ziltener
  • proof of subsitution theorem: Thm 6.6.1
  • characterization of Riemann integrability in terms of set of points of discontinuity
    • 7a. Week 23 (June 5):
  • A remark on step functions as used in Theorem 6.4.2.
  • 6.7: partitions of unity.
  • 6.8: approximation of Riemann integrable functions
  • 6.10: absolute Riemann integrability:
  • Thm 6.10.6 and Cor. 6.10.7: change of variables by absolutely Riemann integrable functions.
    • 7b. Week 23 (June 7): From the extra lecture notes (see blackboard):
  • the notion of density on a linear space
  • reformulation of substitution of variables with densities
  • densities on a submanifold
  • integral of a density over a submanifold
  • the Euclidean density on a submanifold. Calculation in local chart.
    • This covers the equivalent of the following material in the book: vol 2, pp 487 - 499.

    8a. Week 24 (June 12):
  • Example 7.4.9 III Euclidean density of a hypersurface, p 507.
  • 7.5: Thm 5.7.1: domain with $C^k$-boundary
  • Def 7.5.3, Lemma 7.5.4: outer normal to the boundary
  • proof of divergence theorem of Gauss
  • Theorem 7.6.1 with proof
  • partial integration: Corollary 7.6.2
  • skip 7.7
  • 7.8.1: definition of vectorfield
  • Def 7.8.3: definition of divergence
  • Theorem 7.8.5, Def 7.8.6.
  • Study Example 7.9.6 on your own.
    • 8b. Week 24 (June 14): last lecture Analyse in meer variabelen
  • Def 8.1.1: reformulated with unit tangent
  • Def 8.1.6: curl in R2, Def 8.1.9: curl in R3
  • skip section 8.2
  • Proof of Defi 8.3.3, Thm 8.3.5: Green s integral theorem
  • Stokes integral theorem, Theorem 8.4.4
    • 9a. Week 25 (June 19): no lecture, but question and rehearsal session

    9b. Week 25 (June 21): no lecture, possibly question and rehearsal session

    10. Week 26: exam week

    Last change: June 2, 2018