In this course we will 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 basic operations of vector fields, such as "divergence" or "rotation". They will also learn "Green's theorem", "divergence theorem" and "Stokes' theorem".
At the end of this course, students are expected to:
-- be able to calculate inner and outer products
-- be able to calculate line integrals of vector fields along curves
-- be familiar with parametrization of curves and surfaces
-- understand the meaning of gradient, divergence, and rotation, and be able to calculate them
-- understand what Green's theorem means and know how to use it
-- 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
Outer product, vector fields, line integral, gradient, divergence, rotation, Green's theorem on the plane, 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||Curves and surfaces in the space||Details will be provided in class.|
|Class 2||Curves and surfaces in the space||Details will be provided in class.|
|Class 3||scalar fields and gradient vectors||Details will be provided in class.|
|Class 4||Line integrals of vector fields||Details will be provided in class.|
|Class 5||Green's theorem and its application||Details will be provided in class.|
|Class 6||Surface integrals and divergence theorem||Details will be provided in class.|
|Class 7||surface integrals and divergence theorem||Details will be provided in class.|
|Class 8||Parametrization of surfaces and tangent spaces||Details will be provided in class.|
|Class 9||Surface area and surface integrals||Details will be provided in class.|
|Class 10||Gauss' divergence theorem||Details will be provided in class.|
|Class 11||Stokes' theorem||Details will be provided in class.|
|Class 12||Applications of divergence and Stokes' theorems||Details will be provided in class.|
|Class 13||Differential forms, wedge product, exterior derivative||Details will be provided in class.|
|Class 14||Integration of differential forms and generalized Stokes' theorem, comprehension check-up||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 and quizzes. 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.