Fall 2020 MATH 3304
Vector Analysis 向量分析

Syllabus 課程大綱

宣導事項

9/8/2020: Welcome to the class!

教學進度

9/8/2020(星期二): Greetings, Introduction to vector fields.

9/11/2020(星期五): Conservative vector fields and finding potential functions.

9/15/2020(星期二): Conservative vector fields, the Fundamental Theorem of Line Integral, and path dependence.

9/18/2020(星期五): More on path dependence, closedness condition.

9/22/2020(星期二): Equivalence of path dependence and conservativeness.

9/25/2020(星期五): Regular k-surfaces.

9/29/2020(星期二): Parametrizations, general k-surfaces.

10/2/2020(星期五): No Class (中秋節)。

10/6/2020(星期二): Tangent and normal vectors.

10/9/2020(星期五): No Class (台灣國慶日)。

10/13/2020(星期二): Geometry of normal vectors, surface area.

10/16/2020(星期五): Stokes Theorem on R^3.

10/20/2020(星期二): Stokes and Divergence Theorem on R^3.

10/23/2020(星期五): Review of dual spaces.

10/26/2020(星期二): Midterm Exam 1.

10/30/2020(星期五): Multi-linear functions on vector spaces.

11/3/2020(星期二): Differential k-forms.

11/6/2020(星期五): Wedge Products.

11/10/2020(星期二): Exterior derivatives and pullbacks.

11/13/2020(星期五): Pullbacks of k-forms and properties.

11/17/2020(星期二): Finish differential forms.

11/20/2020(星期五): Reviews of Linear algebra.

11/24/2020(星期二): Integration of differenntial forms.

11/27/2020(星期五): Regular k-surfaces with boundary, orientations.

12/1/2020(星期二): More on boundary, Introduction to general Stoke's Theorem.

12/4/2020(星期五): Proof of general Stoke's Theorem for surfaces with global parametrization.

12/8/2020(星期二): Finish the proof of general Stoke's Theorem, proof of classical theorems in elementary calculus.

12/11/2020(星期五): Some consequences of Stoke's Theorem.

12/15/2020(星期二): Midterm Exam 2.

12/18/2020(星期五): Introduction to Maxwell's equations.

12/22/2020(星期二): Physical derivations of Maxwell's equations.

12/25/2020(星期五): Maxwell's equations in terms of differential forms.

12/29/2020(星期二): The geometric and analytic aspects of Maxwell's equations, toward gauge theory.

作業

Homework 1 (截止日 2020/9/18 In Class):
Salas Section 16.9: 8, 10, 15, 28, 29, 30.

Homework 2 (截止日 2020/9/25 In Class):
Salas Section 18.1: 14, 18, 28.
Salas Section 18.2: 3, 6, 17, 20, 26.
Salas Section 18.5: 11, 13, 16, 21, 24, 32.

Homework 3 (截止日 2020/10/23 In Class):
Galbis-Maestre Section 3.5 (p. 105): 2, 6, 9, 10.
Galbis-Maestre Section 4.4 (p. 125): 4, 6, 7, 8.
Salas Section 18.9: 4, 10, 16, 19, 20.
Salas Section 18.10: 6, 8, 10, 12, 17.

Homework 4 (截止日 2020/11/20 In Class):
Galbis-Maestre Exercises (p.183-184) 6.7.1 - 6.7.8.

Homework 5 (截止日 2020/12/11 In Class):
Galbis-Maestre Exercises (p. 267) 8.7.1 - 8.7.3.
Galbis-Maestre Exercises (p. 317-318) 9.7.1 - 9.7.7.

Homework 6 (截止日 2020/1/5 與期末考一併繳交):
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