MATH7712 Differential Geometry

2016

4.5

3 x 50-minute lectures weekly
1 x 50-minute tutorial weekly

1 2 of MATH2701, MATH2711, MATH2014, MATH2111
1a ENGR2711 - Engineering Mathematics
1b 2 of MATH2111, MATH1141
2 Admission into GCEB-Graduate Certificate in Engineering (Biomedical)
2a Admission into GDPEB-Graduate Diploma in Engineering (Biomedical)
2b Admission into MEB-Master of Engineering (Biomedical)
2c Admission into GCESI-Graduate Certificate in Engineering (Smart Instrumentation)
2d Admission into MESI-Master of Engineering (Smart Instrumentation)
2e Admission into GCEE-Graduate Certificate in Engineering (Electronics)
2f Admission into GDPEE-Graduate Diploma in Engineering (Electronics)
2g Admission into MEE-Master of Engineering (Electronics)
2h Admission into HBSC-Bachelor of Science (Honours)
2i Admission into HBIT-Bachelor of Information Technology (Honours)
2j Admission into MSCMT-Master of Science (Mathematics)
Must Satisfy: (((1 or 1a or 1b)) or ((2 or 2a or 2b or 2c or 2d or 2e or 2f or 2g or 2h or 2i or 2j)))

MATH4712 has been successfully completed

Students undertaking the one year honours program should check to make sure they have completed the appropriate background form their undergraduate degree/s.

Curves in plane and space, curvature and torsion of a curve, Frenet formulas; smooth surfaces, tangent plane, smooth functions on surfaces and their differentials, diffeomorphism, first quadratic form of a surface, isometric surfaces, principal curvatures, Gauss and mean curvatures, second quadratic form, Gauss¿ Theorem Egregium, geodesic curves; differentiable manifolds, charts, atlases, tangent vectors, tangent space; Riemannian manifolds, length of a curve on a Riemannian manifold, volume of a Riemannian manifold; vector and convector fields, tensors, affine connections; Riemann curvature tensor, Ricci tensor, scalar curvature; Einstein equations for a gravitational field.

This topic introduces basic differential geometry of lines and surfaces in space and elements Riemannian geometry. This subject is to be considered as an integral part of the general mathematical knowledge.

At the completion of the topic, students are expected to be able to:

1. Have a proper knowledge of the geometry of curves and surfaces in space
2. Have a basic knowledge of the Riemannian geometry and of its applications
3. Better understand the role and significance of mathematics in the modern world