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. |

**Instructor:** Dr. Gantumur Tsogtgerel

**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.

**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% }