The purpose of this module is to show students to how ideas from multivariable calculus and linear algebra can be put together to form an invariant language which is quite useful in understanding the structure of very naturally occurring geometric objects in mathematics.   

Recommended Module: MA0212 Linear Algebra

Comfortably compute various quantities on curves and surfaces (such as frame fields, differentials forms, area, and the Gauss map), understand the notion of curvature through the use of curves and surfaces, and understand the general definition of manifolds and the respective generalizations of curvature through tensors.

27 fifty-minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions for tutorial classes.

Skills:

How to pass from « local » theory (i.e. calculations from multivariable calculus in n-dimensions) to « global » theory, comfortability with the algebraic structure of various geometric objects, ability to use techniques from linear algebra in settings outside of the algebra course sequence.

Transferable Skills:
Ability to understand partial differential equations in the setting of manifolds, algebraic/geometric intuition which can be a base for algebraic topology and algebraic geometry, comfortability in passing between abstract geometric language (topological spaces, groups, forms, etc.) and concrete analytic quantities (charts, arc-length, Gaussian curvature, etc.).

Formative assessment is by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures and tutorial classes.

The summative assessment is the written examination at the end of the module.  .  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of three from four equally weighted questions.

Tools from Multivariate Calculus:

• Tangent vectors
• Differentials of functions
• Inverse function theorem and implicit function theorem

Curves:

• Frenet curves and frames
• Curvature and torsion
• Frenet equations and the fundamental theorem of the local theory of curves
• Winding numbers and rotation index
• Global theory of curves: Fenchel's theorem, Hopf's theorem, Fary-Milnor

Surfaces:

• Surface parametrizations
• First fundamental form
• Gauss map and the second fundamental form
• Curvauture of surfaces
• Theorema Egregium
• Surfaces of rotation
• Rules surfaces
• Minimal surfaces

Time permitting:

• Covariant differentiation
• Geodesics
• Gauss-Bonnet Theorem

Kuhnel, W. 2015. Differential geometry; curves, surfaces, manifolds. 3rd ed. American Mathematical Society.

O'Neill, B. 2006. Elementary Differential Geometry. 2nd ed. Academic Press. 1st ed. available as eBook via http://bit.ly/2uGDnou

do Carmo, M. 2016. Differential Geometry of Curves and Surfaces. 2nd ed. Dover Publications.