Problem sets, solutions

  • Problem set 1 due Thursday September 29
  • Problem set 2 [tex] due Thursday October 20
  • Problem set 3 [tex] due Tuesday November 8
  • Practice midterm, hints, and solutions
  • Solutions to midterm problems
  • Maximization/minimization problems with solutions
  • Problem set 4 [tex] due Tuesday November 22
  • Problem set 5 [tex] due Thursday December 1
  • Practice final
  • Lecture notes

  • Review of single variable calculus
  • Differentiation
  • Introduction to manifolds
  • Review of linear algebra
  • Riemann integral
  • Stokes' theorem
  • Class schedule

  • TR 08:35–09:55, Strathcona Anatomy & Dentistry 1/12

    Date Topics
    T 9/6 Review of continuity. Little "o" notation. Approximation by a constant. Sequential criterion of continuity.
    R 9/8 Differentiability. Carathéodory's definition. Approximation by a linear function. Sequential criterion.
    T 9/13 Continuity and differentiability of vector valued functions.
    R 9/15 Separate and joint continuity.
    T 9/20 More on continuity. Directional and partial derivatives.
    R 9/22 Fréchet derivative. Jacobian matrix.
    T 9/27 Caratheodory's criterion. Chain rule. Second order derivatives.
    R 9/29 Hessian matrix and its symmetricity.
    T 10/4 Quadratic approximation.
    R 10/6 Inverse function theorem for single variable functions.
    T 10/11 Inverse function theorem.
    R 10/13 Inverse function theorem. Implicit function theorem in two dimensions.
    T 10/18 Curves. Tangent line. Tangent vectors.
    R 10/20 Vectors and covectors. Coordinate transformations.
    T 10/25 Manifolds. Implicit function theorem.
    R 10/27 Applications of the implicit function theorem. Implicitly defined manifolds.
    M 10/31 Midterm exam, 18:05-19:05, Stewart Biology S1/4
    T 11/1 The preimage theorem. Level surfaces. The orthogonal group.
    R 11/3 Tangent spaces of manifolds. Critical points. Local extrema.
    T 11/8 Conormal spaces. Lagrange multipliers.
    R 11/10 Bounded and closed sets. Weierstrass' existence theorem.
    T 11/15 Hessian test. The Riemann integral. The fundamental theorem of calculus.
    R 11/17 Fubini's theorem. Negligible sets. Jordan content.
    T 11/22 Linear transformations.
    R 11/24 Change of variables. Oriented curves. Line integrals.
    T 11/29 Oriented curves. Green's theorem. Stokes' theorem.
    R 12/1 Divergence theorem.
    R 12/15 Final exam, 18:00-21:00.

    Reference books

  • Jerrold Marsden and Anthony Tromba, Vector calculus. W. H. Freeman (Any edition).
  • Michael Spivak, Calculus on manifolds. Westview Press 1971.
  • Online resources

  • Dan Sloughter Calculus of several variables
  • James Cook's lecture notes
  • Tom Apostol's calculus books
  • Richard Hammack Book of proof
  • Illustrations

  • Map projections
  • Republic P-47 Thunderbolt by Anders Lejczak
  • Sand Speeder
  • Global weather conditions by Cameron Beccario
  • Blood fluid dynamics in arteries and veins
  • Course outline

    Instructor: Dr. Gantumur Tsogtgerel

  • Office hours: TR 10:00–11:00
  • Office: Burnside Hall 1123

    Prerequisite: MATH 133 (Linear Algebra and Geometry) and MATH 222 (Calculus 3) or consent of Department.

    Restriction: Intended for Honours Mathematics, Physics and Engineering students. Not open to students who have taken or are taking MATH 314.

    Topics: An in-depth study of certain Calculus 3 topics, as well as some new topics, including surfaces, manifolds, vector fields, and integration of vector fields.

  • Differentiation, Taylor's theorem, inverse function theorem, implicit function theorem
  • Curvilinear coordinates, embedded manifolds, gradient, vector fields
  • Critical points, Hessian test, Lagrange multipliers, Weiertsrass existence theorem
  • Riemann integral, Fubini's theorem, partition of unity, integration over manifolds
  • Divergence, rotor, curl, Gauss-Ostrogradsky theorem, Green’s theorem, Stokes theorem

    Calendar description: Partial derivatives; implicit functions; Jacobians; maxima and minima; Lagrange multipliers. Scalar and vector fields; orthogonal curvilinear coordinates. Multiple integrals; arc length, volume and surface area. Line integrals; Green's theorem; the divergence theorem. Stokes' theorem; irrotational and solenoidal fields; applications.

    Homework: 4-5 written assignments.

    Exams: A midterm and final.

    Grading: Homework 20% + MAX{ Midterm 20% + Final 60% , Final 80% }

    Midterm exam

  • The midterm exam is on Monday October 31, in Stewart Biology S1/4, 18:05-19:05.
  • It is a closed book, closed note exam. No calculators will be allowed or needed.
  • Practice problems
  • In case you are stuck, here are some hints.
  • Solutions to most of the practice problems
  • The first 3 problems from Problem set 3 may be used as additional practice problems.