This course is a succession of "Introduction to Analysis III" in the third quarter. We will continue to teach "vector calculus", that is a calculus for scalar fields (single-valued functions) and vector fields (multivalued functions) . Each lecture will be followed by a recitation (a problem-solving session).
The students will learn "divergence theorem" and "Stokes' theorem" on surface integrals. They will also learn differential forms to formalize these theorems in a unified manner, as extensions of the "fundamental theorem of calculus".
At the end of this course, students are expected to:
-- understand the tangent vectors and tangent space of surfaces
-- be able to calculate surface integrals of vector fields
-- understand the meaning of divergence theorem and Stokes' theorem
-- be able to calculate differential forms
tangent vector, surface integral, divergence theorem, Stokes theorem,
differential forms, exterior derivative
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
This is a standard lecture course with recitation sessions. Homework will be assigned every week. There will be occasional quizzes.
Course schedule | Required learning | |
---|---|---|
Class 1 | Parametrization of surfaces and tangent spaces | Details will be provided in class. |
Class 2 | Recitation | Details will be provided in class. |
Class 3 | Surface area and surface integrals | Details will be provided in class. |
Class 4 | Recitation | Details will be provided in class. |
Class 5 | Gauss' divergence theorem | Details will be provided in class. |
Class 6 | Recitation | Details will be provided in class. |
Class 7 | Stokes' theorem | Details will be provided in class. |
Class 8 | Recitation | Details will be provided in class. |
Class 9 | Applications of divergence and Stokes' theorems | Details will be provided in class. |
Class 10 | Recitation | Details will be provided in class. |
Class 11 | Poisson's equation | Details will be provided in class. |
Class 12 | Recitation | Details will be provided in class. |
Class 13 | Differentia forms, wedge product, exterior derivative | Details will be provided in class. |
Class 14 | Recitation | Details will be provided in class. |
Class 15 | Integration of differential forms and generalized Stokes' theorem, quiz | Details will be provided in class. |
None required
None required
Final exam 50%, assignments and quizzes 50%.
Students are expected to have passed
-- Calculus (I/II), Linear Algebra (I/II), and their recitations.
-- Introduction to Analysis I/II.