Instructor Dragos Oprea
Lectures: MWF, 12pm-12:50pm, APM B-412.
Christian Woods

This course is the second in the Mathematics Department's Honors sequence. It covers multivariable differentiation: point-set topology, limits and continuity, derivatives in several variables, tangent spaces, Taylor polynomials, Min/max in several variables, Lagrange multipliers, manifolds.

The three courses in the Honors sequence, Math 31ABC, cover essentially the same material as do Math 20F, 20C, and 20E, respectively, but at a more sophisticated conceptual level. The Honors sequence emphasizes proofs, so students completing it will be exempt from taking Math 109 (Mathematical Reasoning). The Honors sequence is intended for mathematics majors and prospective mathematics majors (although others are very welcome) and gives a much better view of what upper-division mathematics is like than does the standard Math 20 sequence.

Prerequisities: A grade of B- or better in Math 31AH.
The grade is computed as the following weighed average:

  • Homework 20%, Midterm I 20%, Midterm II 20%, Final Exam 40%.
  • Textbook:J. H. Hubbard and B. B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, Fourth Edition.

    Reading the sections of the textbook corresponding to the assigned homework exercises is considered part of the homework assignment. You are responsible for material in the assigned reading whether or not it is discussed in the lecture. It will be expected that you read the assigned material in advance of each lecture.


    Homework problems will be assigned on the course homework page. There will be 8 problem sets, due certain Fridays at 4:30PM in the TA's mailbox. The tentative due dates are January 10, 17, 24, February 7, 14, 21, March 5, 14.

    You may work together with your classmates on your homework and/or ask the TA (or myself) for help on assigned homework problems. However, the work you turn in must be your own. No late homework assignments will be accepted.

    There will be two midterm exams given on January 31 and February 28. There will be no makeup exams.

    The final examination will be held on Wednesday, March 19, 11:30-2:30. There is no make up final examination. It is your responsability to ensure that you do not have a schedule conflict during the final examination; you should not enroll in this class if you cannot sit for the final examination at its scheduled time.

    Announcements and Dates
    • Monday, January 6: First lecture
    • Monday, January 20: MLK Day. No class
    • Friday, January 31: Midterm I
    • Monday, February 17: Presidents' Day. No class
    • Friday, February 28: Midterm II
    • Friday, March 14: Last Lecture
    • Wednesday, March 19 : FINAL EXAM, 11:30-2:30
    Lecture Summaries
    • Lecture 1: Course overview. Notions of point-set topology. Open sets. Closed sets. Interior. Closure. Boundary. Examples.
    • Lecture 2: Limits of sequences. epsilon-delta definitions. Limits of functions. Properties. Examples.
    • Lecture 3: Continuity via epsilon-delta's. Properties of continuous functions. Examples.
    • Lecture 4: Linear transformations are continuous. Continuity and open/closed sets. Open sets of matrices.
    • Lecture 5: Uniform continuity and examples. Compact sets. Sequentially compact sets. Properties and examples.
    • Lecture 6: Images of compact sets under continuous maps are compact. Inf and sup, min and max. Continuous functions over compact sets achieve min and max.
    • Lecture 7: Min and max via derivatives. Mean value theorem and proof. Applications of the mean value theorem: monotonic functions via sign of derivatives. Functions of several variables.
    • Lecture 8: Graphs, level contours, cross sections. Paraboloids, saddles, cones. Partial derivatives.
    • Lecture 9: Partial derivatives and tangent plane to graphs. Jacobian matrix. Total derivative. Comparison between the total derivative and the Jacobian. Differentiability versur continuity.
    • Lecture 10: Directional derivative. Total derivative and the directional derivative. Direction of steepest increase.
    • Lecture 11: Pathological functions. Function which admits partial derivatives but not a total derivative. Functions of class C^1. Mean value theorem in several variables.
    • Lecture 12: Rules for calculating derivatives: sums, products, quotients, composition. Chain rule.
    • Lecture 13: Examples of the chain rules in matrix form and partial derivative form. Polar coordinates. Gradient is perpendicular to level sets.
    • Lecture 14: Tangent planes to level sets. Second order derivatives. Functions of class C^k, C^{\infty}. Equality of mixed partial derivatives.
    • Lecture 15: Example of a function for which the order of differentiation matters. Taylor polynomials and functions of class C^{omega}.
    • Lecture 16: Taylor polynomials in several variables. Examples of Taylor polynomials. Shortcuts to finding Taylor polynomials.
    • Lecture 17: Critical points. Second derivative test. Saddle points, local min and local max.
    • Lecture 18: Min and max of functions on compact sets: example when the boundary is a rectangle. More general boundaries and Lagrange multipliers.
    • Lecture 19: Examples of Lagrange multipliers. One or several constraints.
    • Lecture 20: Introduction to the implicit function theorem. Motivation and statement.
    • Lecture 21: More on the implicit function theorem. Examples. The solution set looks like the graph of a function. The calculation of the derivative of the implicit function.
    • Lecture 22: The inverse function theorem. Locally invertible functions. Examples. Complex exponential and logarithm.
    • Lecture 23: Smooth manifolds. Examples from the implicit function theorem. Parametrization of manifolds. Graphs are naturally parametrized.
    • Lecture 24: Reparametrization. Standard tangent planes to manifolds. Examples: tangent space of manifolds given by equations, tangent space of manifolds given by parametrization.
    • Lecture 25: More examples of tangent spaces. Critical points of functions over manifolds. Critical points over parametrized manifolds.
    • Lecture 26: Critical points over manifolds given by equations. Proof of Lagrange multipliers with several constraints.