# 幾何學(二)
在幾何學(二)中，我們學習跟肥皂泡 (soap film or soap bubble) 相關的數學理論--minimal surface theory。課本選用 [John Oprea: *The Mathematics of Soap Films: Explorations with Maple*](https://bookstore.ams.org/stml-10) 。

From the Preface of the book:

* This book is about the mathematics which describes the geometric properties of soap films.
* Scientific principles are often reflected in geometry.

* Using idea from plane geometry, differential geometry, complex analysis and the calculus of variations, we can begin to understand why soap films take the shapes they do.
* As we shall see, the effects of surface tension lead a soap film to minimize its surface area.
* This allows us to study soap film shapes from a purely mathematical viewpoint.

預備知識: Calculus, Ordinary differential equations. The relevant notions from differential geometry and complex variables will be discussed when needed.

# Schedule
Week 1-2: Geometry of curves and surfaces.
| Week | Topics |Reading |
| -------- | -------- | -------- |
| 1 | parametrized surfaces; normal curvature | 2.1, 2.2; do Carmo 3.2 |
|2|first and second fundamental forms; mean curvature; review of vector analysis; divergence theorem on surfaces|do Carmo 2.5, 3.3; any Calculus textbook|
|3|Physics of soap films, Laplace-Young equation, Plateau's Rules|1.2, 1.4, 1.5|
|4|Plateau's Rules; Gauss curvature; 陳健綸 Steiner 3-point problem|1.5, 2.5|
|5|The Basics of Minimal Surfaces|3.2; do Carmo 3.5|
|6|Area Minimization and Soap Films; Isothermal Parameters|3.3, 3.4; do Carmo 3.5|
|7|Harmonic Functions and Minimal Surfaces; Complex Variables|3.5, 2.4|
|8|The Weierstrass-Enneper Representations|3.6|
|9|The Gauss Map, Stereographic Projection|3.7, 3.8|
|10|Creating Minimal Surfaces from Curves|3.9|
|11|To Be or Not To Be Area Minimizing|3.10|
|12|Constant Mean Curvature|3.11|
|13|Capillary||
|14||4.1, 4.2, 4.3|
|15||4.4|
|16||4.5|
|17|Student presentations||
|18|Student presentations||

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[Course Notes](https://www.dropbox.com/s/ctpmasxqov2z6ut/Outline%20of%20Geometry%202.pdf?dl=0)

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# Links
[表面張力的現象與成因](http://highscope.ch.ntu.edu.tw/wordpress/?p=1529)
[Surface tension (Khan Academy)](https://www.khanacademy.org/science/biology/water-acids-and-bases/cohesion-and-adhesion/v/surface-tension)
[Du Nouy ring method for surface tension measurement](https://www.youtube.com/watch?v=N-9yeSxdEkM)
[Marangoni effect](https://en.wikipedia.org/wiki/Marangoni_effect)
###### tags: `past course`