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.

MAT 116: Differential Geometry
Approved: 2003-03-01 (revised 2013-01-01, B. Temple)

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Elements of Differential Geometry, 1st Edition by Richard Millman and George Parker; Pearson Publishing; $40.00-73.00 via Amazon Books. Alternate texts Differential Geometry of Curves and Surfaces, 1st Edition by Manfredo P. Do Carmo; Pearson Publishing;
Search by ISBN on Amazon: 978-0132641432

Prerequisites:

Completion of course MAT 125A.

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:

MAT116 introduces the concepts of Riemannian geometry within the context of the theory of two dimensional surfaces. Differential geometry gives a coordinate independent meaning to the vector calculus taught in MAT21D, and is the foundation for the abstract theory of manifolds. MAT116 introduces the language of tensor analysis used in engineering, and in describing the theory of curved surfaces, provides an introduction to the modern notions of connection, co-variant derivative and ultimately the Riemann curvature tensor.

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:

To assess the learning outcome for this course, there will be homework, midterms, and one final exam, together with participation during lecture.