Jay Gopalakrishnan

Neuberger Hall 364

Tue, Thu: 10:00 - 11:50am

• Did you like calculus?
If so, you might enjoy this course. (If you struggled with MTH 254, this course is not for you.)



In calculus, you differentiated along 1-dimensional curves. Way cool!	You also analyzed 2-dimensional surfaces. Super!	You even integrated over 3-dimensional solids? You are a master!

Now take it to the next level:


• Wouldn't you like to know how to integrate over curved n-dimensional objects?
If so, take this course and learn about infinitesimal volumes on n-manifolds for any integer dimension $n \ge 1$, even beyond the visualizable 3 dimensions.

In calculus, you learnt to compute all these integrals: \[ \begin{aligned} \int f \; dx, && \int_C \vec{F} \cdot d\vec{t}, && \iint_D f \, d S, && \oint_{\partial D} P\, dx + Q\, dy, \\ \int_a^b f\; dx, && \oint_{\partial D} \vec{F} \cdot d\vec{t}, && \iint_S \vec F \cdot d \vec S, && \iiint_E f \; d V, \quad\ldots \end{aligned} \]

• Would you like to know a way to unify all these integrals into one?
If so, come and learn about integrating forms over manifolds.

The goal of this course is to help students acquire facility with calculations using differential forms and integrals over manifolds in any dimension. The focus is not on proof writing, but rather on calculational techniques.

MTH 254, MTH 261.

The learning material will not come a single text, so it will be particularly important to attend the lectures. You are not required to buy any textbook for this course. Some materials covered in the lectures can be found in references like these:

• Differential Forms with Applications to the Physical Sciences by Harley Flanders, 1989
• Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus by Michael Spivak, 1971.
• Vector Calculus, Linear Algebra, and Differential Forms - A unified approach 4th Edition, by John Hubbard & Barbara Hubbard, 2009.

The last book will be on reserve for this course in the Millar Library. All other course materials, including homeworks, and this page, are accessible from the D2L page for the course. Students are required to make sure that they receive emails sent to their D2L accounts.

In the first few lectures, the students will make the transition, in notation and concepts, from earlier multivariable calculus courses like MTH 254 to those continually used in this course. Then, viewing curves, surfaces, and solids as instances of general "manifolds", we generalize the integrals over curves and surfaces to a general integral over k-manifolds in n-dimensions.

Differential forms are introduced early. By the end of the course, the "Stokes theorem" will be learnt as one statement that encompasses and generalizes multiple theorems in MTH 254. Time permitting, we conclude by discussing electromagnetism in the language of forms.

Caitlin Graff   [cgraff @ pdx.edu]

Instructor is available Tuesdays 2pm-3pm (in NH 309) or by appointment.

Homework will be given throughout the course. Students are not required to turn in solutions. The entire grade is determined by quizzes held in class. (There are no midterm or final examinations.) The quizzes will be based on the homework and their dates will be announced in class.