Theory Analyse in meer variabelen (WISB 243)

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Analyse in meer variabelen, period 4

All references are to the books by Duistermaat and Kolk

1a. Week 17 (April 26):
  • 2.1: linear mappings and matrices, distinction between notions, norm
  • 2.2: differentiability, definition 2.2.2, criterion prop 2.3.2,
  • 2.4: chain rule
  • 2.7: higher order derivatives and partial derivatives, notion of C^p functions
  • 2.8: Taylor's formula in several variables, Thm 2.8.3, multi-index notation, integral form of remainder term.
  • 2.9: critical points and Hessian (advice: read on your own, this material has been discussed in the course Introduction to Analysis in Several Variables.
    • 1b. Week 17 (April 27): King's day, no lecture.
    2a. Week 18 (May 3):
  • 2.8 completion of Taylor in several variables
  • 3.1 notion diffeomorphisms
  • 3.2 formulation inverse function theorems, both the local and global version (for proof, see course Introduction to Analysis in Several Variables).
  • 3.4 implicitly defined mappings
  • 3.5 implicit function theorem with proof based on inverse function theorem
    • 2b. Week 18 (May 4):
  • 3.5: finish proof of implicit function theorem: formula for the derivative, determinant of block triangular matrix
  • 3.6: application A: simple zeroes behave smoothly with coefficients
  • 3.2: proposition 3.2.9 the C^k inverse and implicit function theorems
  • 3.7: skipped
  • 4.1, 4.2: beginning of submanifold theory, the implicit function theorem and the zero set of a function represented as a graph.
    • 3a. Week 19 (May 10):
  • 4.1, 4.2: implicit function theorem and zero set given as graph
  • def. 4.2.1 replaced by definition 10 in extra notes (see blackboard, course information)
  • Lemma 11: equivalence of the definitionss
  • Definition 4.2.4: immersion, embedding, coordination and chart
  • Def 4.2.6: submersion
  • 4.3: Immersion theorem, see Lecture Notes, Thm. 12.
    • 3b. Week 19 (May 11):
  • 4.3 Cor. 4.3.2: image of embedding is submanifold Lemma 4.3.3: coordinate change: postponed to next lecture
  • 4.4 Examples: read on your own
  • 4.5 Submersion theorem, see Lecture Notes, Thm. 13.
  • 4.6 Examples: read on your own
  • 4.7 equivalent characterizations of a manifold, see also lecture notes, pages 9,10.
  • 4.8 skipped
  • 5: tangent space: definition tangent vector, behavior under diffeomorphism
    • 4a. Week 20 (May 17):
  • 5.1: linearity tangent space
  • 5.1: relation of tangent space to submanifold characterizations
  • Lemma 4.3.3 coordinate change (read proof on your own) and general definition of manifold
  • 5.2: Tangent mapping
  • 5.3: examples of tangent space calculations
  • 5.4: Lagrange multipliers from the viewpoint of geometry (read on your own: extra lecture notes, p. 16)
  • rest of chapter 5: skipped
    • 4b. Week 20 (May 18):
  • 6: integration
  • 6.1 Rectangles def 6.1.1, partition, Volume, Prop. 6.1.2, refinement, Lemma 6.1.3.
  • 6.2: Riemann integrability in several variables: lower sum and upper sum, Def 6.2.1,
  • behavior under refinement: 6.2.2, Def 6.1.3 of upper and lower integral, Def 6.2.4 and Prop 6.2.5
  • Def 6.2.6: support (drager) of a function, Def 6.2.7, Thm 6.2.8.
  • 6.3: the notion of Jordan measurability: Def 6.3.1, Thm 6.3.2, Cor 6.3.3,
    • 5a. Week 21 (May 24): Pentecost, no lecture
    5b. Week 21 (May 25):
  • Def 6.3.4, Thm 6.3.5, see also Extra Lecture Notes, Lemma 45, Thm 46,
  • Def 6.3.6, Cor 6.3.7, Cor. 6.3.8
  • 6.4: successive integration, ELN: stepfunctions, Lemma 48, Book 2: Thm 6.4.2, Cor 6.4.3, Thm 6.4.5.
    • 6a. Week 22 (May 31):
  • Finish 6.4: successive integration
  • Thm 6.6.1: Formulation of change of variables theorem for compactly supported functions.
  • Cor. 6.6.2: transformation of volume and geometric interpretation of change of variables
  • Example 6.6.4: polar coordinates
  • 6.10: absolute Riemann integrability, local integrability, improper Riemann integral and extension of change of variables: 6.10 and 6.10.2 with avoidance of partition of unity, Lemma 6.10.13, Def 6.10.5, Th. 6.10.6, Cor. 6.10.7. Ex. 6.10.8.
    • 6b. Week 22 (June 1):
  • Proof of the substitution of variables theorem for compactly supported functions
  • 6.7: compactness, Heine-Borel (see ELN)
  • 6.8 approximation with compactly supported continuous functions and reduction to local version of substitution of variables (SOV).
  • proof of local version of SOV with implicit function theorem: 6.9.
    • 7a. Week 23 (June 7):
  • volume of a parallellepiped, book Example 6.6.3
  • partitions of unity, book and ELN: p. 32
  • absolute convergence of Riemann integrals book, section 6.10
  • ELN (Extra lecture notes): densities on linear spaces , Def 52, Lemmas 53,54,55, 56, densities on open set
  • ELN: Theorem 57, alternative formulation of substitution of variables
    • 7b. Week 23 (June 8):
  • Densities on a submanifold, ELN: Def 58, Lemma 59, Lemma 60, Remark 61, Def 62, Lemmas 63 - 64
  • Integration of density on a submanifold: ELN Prop 65, Def 66.
  • Euclidean density on a submanifold, Lemma 68, Def 69, Lemma 70, Lemma 71
  • mentioned: Corollary 89 and surface integrals in R3.
    • 8a. Week 24 (June 14):
  • Several formulas for Euclidean density ELN: Cor 72, 73, Def 74 of unit normal, Lemma 75, Lemma 76,
  • Cor. 77, Lemma 78, Cor. 79, Rem 80, Lemma 81.
  • Partial integration and Gauss' theorem, see ELN pp 46 - 50.
    • 8b. Week 24 (June 15):
  • completion partial integration and Gauss.
    • 9a. Week 25 (June 21): No lecture: Final opportunity to ask questions
    9b. Week 25 (June 22):
    10. Week 26 (June 28): exam week.


    Last change: March 26, 2021