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||Poisson's equation||Details will be provided in class.|
|Class 10||Recitation||Details will be provided in class.|
|Class 11||Differentia forms, wedge product, exterior derivative||Details will be provided in class.|
|Class 12||Recitation||Details will be provided in class.|
|Class 13||Integration of differential forms and generalized Stokes' theorem||Details will be provided in class.|
|Class 14||Recitation||Details will be provided in class.|
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Based on the final exam, quizzes, and the problem solving situation in the recitation sessions. Details will be provided in the class.
Students are expected to have passed
-- Calculus (I/II), Linear Algebra (I/II), and their recitations.
-- Introduction to Analysis I/II.