## 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

The 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: Wed 10-12

### Lectures

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

First lecture on Monday, 6 July 2020

### Tutorials

Mon at 3 in MA 241

### Internal Assessment

10 weekly assignments

**Attention**: There is no plussage. The assignments count for **30%** of the final grade.

### Final mark

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

**F = 0.7E + 0.3A**

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 integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**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 students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.

### 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 $$r=4a\cos^3rac{ heta}3$$ 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?