## MATH306 Geometry of Curves and Surfaces

Second Semester |

### Paper details

This paper is an introduction to differential geometry; its focus is the structure of two-dimensional surfaces.

### Potential students

This paper is particularly relevant to Mathematics and Physics majors.

### Prerequisites

MATH 202 (Linear Algebra), MATH 203 (Calculus of Several Variables), COMO204 is highly recommended

### Course Outline

The paper will cover the following topics:

- Curves in the plane and in space (parametrised curves, arc length, Frenet-Serret equations)
- Regular Surfaces (regular values, functions on surfaces, first and second fundamental forms)
- Intrinsic geometry of surfaces (the Gauss theorem, parallel transport and geodesics, Gauss Bonnet and applications)

### Required Text

Andrew Pressley, Elementary Differential Geometry, Springer Verlag

(this book is freely downloadable in electronic form from Springer Verlag)

### Lecturer

Prof Jörg Frauendiener, room 223

Office hours: Tue 10-12

### Lectures

3 per week: Mon, Wed and Fri at 9 in MA 241

First lecture on Wednesday, 12 July 2017

### Tutorials

Mon at 3 in MA 241

### Internal Assessment

10 weekly assignments

**Attention**: There is no plussage. The assignments count for **40%** of the final grade.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = 0.6E + 0.4A**

where:

- E is the Exam mark
- A is the Assignments mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic endeavours at the University of Otago are built upon an essential commitment to academic integrity.**

The two most common forms of academic misconduct are *plagiarism* and *unauthorised collaboration*.

#### Academic misconduct: **Plagiarism**

Plagiarism is defined as:

- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

#### Academic misconduct: **Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.

### Johann Carl Friedrich Gauss...

..., 1777-1855, is well known for his contributions to applied mathematics and physics as well as to pure mathematics. His potential was recognized at the age of 7 when he amazed his teacher by summing the integers from 1 to 100 instantly by spotting that the sum was 50 pairs of numbers each pair summing to 101. Gauss had a major interest in differential geometry and published many papers on the subject, e.g. his most renowned work in this field*Disquisitiones generales circa superficies curva*(1828), which arose from his geodesic interests but contained geometrical ideas such as Gaussian curvature. Much of his work has a distinctly practical flavour.

### Cayley’s sextic

This is Cayley’s sextic, with polar equation $$r=4a\cos^3\frac{\theta}3$$ and its corresponding evolute (the locus of its centre of curvature).

### Sample problem

To prove that in equal intervals of time a planet sweeps through equal areas with respect to the sun.*The only force on a planet is towards the sun, so force (and hence acceleration) perpendicular to this is zero. This acceleration can be expressed in polar coordinates as: $$r\ddot{\theta}+2\dot{r}\dot{\theta}$$ Putting this equal to zero is equivalent to $\frac{d}{dt}(r^2\dot{\theta})=0$ or $r^2\dot{\theta}=\text{ constant}$. The area swept out in a constant interval of time $h$ (say, from $t$ to $t+h$) is $$\int_t^{t+h}\frac12r^2\;d\theta\\= \int_t^{t+h}\frac12r^2\dot{\theta}\;dt\\= \text{constant }\times h$$ i.e. a constant.

* This is one of Kepler's three laws of planetary motion.

**Johannes Kepler**(1571-1630), son of a saloon-keeper and assistant to the Dutch astronomer Tycho Brahe. His three laws were based on empirical data and challenged the purist view that all orbits were circular.

### Sample problem

This pair of intersecting paraboloids traps a volume with an elliptical perimeter. What is the volume trapped?