Instructor Dragos Oprea
Lectures: MWF, 12pm-12:50pm, APM B-412.
Gautam Wilkins

This course is the third in the Mathematics Department's Honors sequence. It covers multivariable integration: Fubini's theorem, change of variables formula, differential forms, exterior derivative, generalized Stokes' theorem, conservative vector fields, potentials.

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 31BH.
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.

    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 April 23 and May 23. There will be no makeup exams.

    The final examination will be held on Wednesday, June 11, 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, March 31: First lecture
    • Wednesday, April 23: Midterm I
    • Friday, May 23: Midterm II
    • Monday, May 26: Memorial Day. No class
    • Friday, June 6: Last Lecture
    • Wednesday, June 11 : FINAL EXAM, 11:30-2:30
    Lecture Summaries
    • Lecture 1: Introduction. Integrating functions over R^n. Dyadic cubes. Lower and upper Riemann sums. Lower and upper Riemann integrals. Integrable functions.
    • Lecture 2: Examples of integrals calculated via Riemann sums. Properties of the integral. Geometric interpretation of the integral of a continuous function. Volume of sets. Sets of volume zero.
    • Lecture 3: Two integrable functions which differ only on a set of volume zero have the same integral. Sets of measure zero. Definition and examples. Riemann integrability theorem.
    • Lecture 4: Cavalieri's principle and volume by slicing. Volume of the ball in R^n.
    • Lecture 5: Fubini's theorem and iterated integrals. Separation of variables for integrals. Switching order of integration. Examples of double integrals.
    • Lecture 6: Setting up the limits of integration in R^3 and examples.
    • Lecture 7: Review of determinants. Linear change of coordinates and proof of the change of variables formula. Volume of the ellipsoid. Arbitrary change of coordinates.
    • Lecture 8: Polar coordinates. Cylindrical coordinates. Spherical coordinates. Examples.
    • Lecture 9: More examples of coordinate changes. What lies ahead: integration over manifolds.
    • Lecture 10: Strategy for integrating over manifolds. Volume of k-dimensional parallelograms in R^n. Parametrizing manifolds and issues with the old conventions about parametrizations.
    • Lecture 11: Relaxed parametrizations. Graphs. Surfaces of revolution such as the torus. Setting up the volume of k-dimensional manifolds.
    • Lecture 12: Examples of volumes of manifolds. Arclength of parametrized curves and the astroid. Surface areas of graphs and the paraboloid. Surface area of surfaces of revolution and the torus.
    • Lecture 13: Area in spherical and cylindrical coordinates. k-forms in R^n as multilinear and alternating functions of k-vectors. The standard k-forms.
    • Lecture 14: All forms can be expressed in terms of the elementary forms. Wedge product and examples. Properties of wedge product. Forms with non-constant coefficients and their values on parallelograms.
    • Lecture 15: Integration of forms over parametrized manifolds as the integral of the pullback of the form under the parametrization. Geometric definition of pullbacks in terms of parallelograms, and method of calculation.
    • Lecture 16: The need of orientations for integration of forms. Orientations of R^n and of subspaces of R^n. Examples. Orientation of manifolds.
    • Lecture 17: Orienting manifolds via nonvanishing top forms. Constructing non-vanishing top forms from normal vectors. Orientation of manifolds defined by global equations. Generalization of the right hand rule.
    • Lecture 18: Summary of orientations. Non orientable manifolds. Integrating forms over oriented manifolds. Orientation preserving parametrizations. Examples.
    • Lecture 19: Forms in R^3 and physical meaning: vector fields, work, flux, mass forms. The gravitational field and its work form, path independence.
    • Lecture 20: Flux in cylindrical and spherical coordinates, examples. Physical meaning of forms in R^n. Faraday 2-form in R^4.
    • Lecture 21: Exterior derivative of forms. Exterior derivative and wedge products. Square of the exterior derivative.
    • Lecture 22: Exterior derivative of functions, work forms and flux forms. Gradient, curl, divergence and identities connecting them. Example: the gravitational field has zero curl and divergence. Gravitational potential.
    • Lecture 23: Pieces of manifolds and their boundary. Boundary orientations. Discussion for dimension 1, 2, 3, and the general case. Orientation of parallelograms and of their boundary.
    • Lecture 24: Generalized Stokes. Exact forms integrate to zero over closed manifolds. Proof of generalized Stokes. Checking the assumptions in Stokes is important: the flux of the gravitational field through the sphere.
    • Lecture 25: Applications of Stokes' theorem. (i) Fundamental theorem of calculus for line integrals. Path independence. Zero-curl fields and potentials. (ii) Green's theorem for work and flux forms.
    • Lecture 26: Applications of Stokes' theorem. (iii) Classical Stokes' theorem in R^3 and examples. (iv) Divergence theorem and examples. Flux of the gravitational field over an arbitrary surface; isolating the singularity.
    • Lecture 27: Physical interpretation of divergence and curl. Review.