# MA134 Geometry and Motion

**Lecturer:** Siri Chongchitnan and Thomas Hudson

**Term(s):** Term 2

**Status for Mathematics students: **Core for Maths

**Commitment:** 30 lectures

**Assessment:** 85% Examination, 15% Assessment

**Formal registration prerequisites: **None

**Assumed knowledge: **Vector identities involving dot and cross products, equations of circles, ellipses and hyperbolae in ℝ^{2}, polar coordinates and sketching polar curves, equations of lines and planes in ℝ^{3}, differentiation and integration techniques, Taylor expansion.

**Useful background:** Parametric curves in ℝ^{2}, Cartesian equations of simple surfaces (e.g. a sphere), plotting curves and surfaces in ℝ^{2} and ℝ^{3} online (e.g. math3d.org) or using Python

**Synergies: **MA133 Differential Equations and all Term 2 MA modules

**Leads To: **The following modules have this module listed as **assumed knowledge** or **useful background:**

- MA254 Theory of ODEs
- MA243 Geometry
- MA250 Introduction to Partial Differential Equations
- MA259 Multivariable Calculus
- MA269 Asymptotics and Integral Transforms
- MA4J1 Continuum Mechanics

This module leads on directly to MA259 Multivariable Calculus and, together with MA133 Differential Equations, provides the foundations for most future applied mathematics modules including MA250 Partial Differential Equations, MA209 Variational Principles. The geometric aspects of the module also lead on to MA3D9 Geometry of Curves and Surfaces. The proper theory of integration of functions of several variables is done in MA359 Measure Theory.

**Content**: When a particle moves in space, it traces out a curve as a function of time. This parametric description of a curve gives us an important connection between geometry and motion.

We will study parametric curves using the tools of *vector calculus*. We will also see how surfaces can be described parametrically using 2 parameters. The properties of curves, surfaces and volumes will be studied using partial differentiation and multiple integrals.

**Aims**: This module aims to indicate to students how intuitive geometric and physical concepts such as length, area, volume, curvature, mass, circulation and flux can be translated into mathematical formulas. It also aims to teach the practical calculation of these formulas and their application to elementary problems in particle/fluid mechanics.

**Objectives**: On successful completion of this module students should be able to:

- Parametrise simple curves and surfaces in cartesian and other coordinates, including polar, cylindrical and spherical coordinates
- Calculate lengths and curvatures of curves in R^3 and demonstrate that length is independent of parametrisation
- Understand and be able to calculate line, surface and volume integrals with respect to various coordinate systems. This includes change of variables and change of order of integration in repeated integrals
- Understand and prove simple properties of a conservative vector field
- State the Divergence and Stokes' Theorems and use them to aid calculations
- Apply all these techniques to problems in mechanics (mass, work, circulation and flux) and geometry (area, volume, centre of mass).

**Books**:

See the reading list on Talis.