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Unit of study_

MATH4068: Differential Geometry

This unit is an introduction to Differential Geometry, one of the core pillars of modern mathematics. Using ideas from calculus of several variables, we develop the mathematical theory of geometrical objects such as curves, surfaces and their higher-dimensional analogues. For students, this provides the first taste of the investigation on the deep relation between geometry and topology of mathematical objects, highlighted in the classic Gauss-Bonnet Theorem. Differential geometry also plays an important part in both classical and modern theoretical physics. The unit aims to develop geometrical ideas such as curvature in the context of curves and surfaces in space, leading to the famous Gauss-Bonnet formula relating the curvature and topology of a surface. A second aim is to remind the students about all the content covered in the mathematical units for previous years, most importantly the key ideas in vector calculus, along with some applications. It also helps to prepare the students for honours courses like Riemannian Geometry. By doing this unit you will further appreciate the beauty of mathematics which originated from the need to solve practical problems, develop skills in understanding the geometry of the surrounding environment, prepare yourself for future study or the workplace by developing advanced critical thinking skills and gain a deep understanding of the underlying rules of the Universe.

Code MATH4068
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
(A mark of 65 or greater in 12cp of MATH2XXX units of study) or [12cp from (MATH3061 or MATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3961 or MATH3962 or MATH3963 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979)]
Assumed knowledge:
Vector calculus, differential equations and real analysis, for example MATH2X21 and MATH2X23

At the completion of this unit, you should be able to:

  • LO1. demonstrate knowledge and understanding of fundamental concepts and theorems in differential geometry.
  • LO2. apply fundamental theorems and concepts of differential geometry in order to solve geometric problems
  • LO3. understand and apply geometric concepts to analyse examples to draw conclusions
  • LO4. evaluate geometric quantities such as torsion and curvature
  • LO5. Synthesise knowledge across a range of topics and write novel mathematical proofs.