## Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

**Approved:**2003-03-01 (revised 2013-01-01, B. Temple)

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

Search by ISBN on Amazon: 978-0132641432

**Prerequisites:**

**Suggested Schedule:**

Lecture(s) |
Sections |
Comments/Topics |

1-2 |
Chapter 1 |
Review of vector algebra and elementary vector analysis |

3-10 |
Chapter 2.1-2.5 |
Local curve theory, Frenet-Serret formulas and applications |

11-25 |
Chapter 4.1, 4.2, 4.3, 4.4, 4.5, 4.7, 4.8 |
Local surface theory, Geodesics, Gaussian curvature. |

As time allows |
Chapter 6 Or Chapter 3 Or |
Gauss-Bonnet Theorem or Isoperimetric inequalities or Introduction to manifolds, tensors, convention/ covariance/contravariance |

**Learning Goals:**

The class begins with the study of curves and their tangent vectors, the first goal being to obtain the coordinate independent notion of a vector based on how components transform under change of coordinates. Following this, the transformation rules for metrics and linear transformations are derived, thereby establishing the notion of contravariant and covariant indices for tensors. The tensor transformation rules are organized by the Einstein summation convention. The class is then devoted to constructing the tensorial expression of the covariant derivative, second fundamental form and Guassian curvature of a surface. The final act is a discussion of Gauss’s Theorem Egregium, his proof that the curvature of a surface can be expressed in terms of metrical properties of the surface alone— determined by measurements confined to within the surface itself, independent of how it is situated within the larger space.

This course is an entry point to advance mathematics. Mastery of this course enhances analytic and problem solving skills of students. At the same time the course supports the ability to design mathematical models in a wide range of applied fields (e.g., this course is the starting point for the mathematics behind Einstein’s theory of general relativity and modern particle physics). This course also offers relevant historical background for the development of 21st century science.

**Assessment:**