PC4230 Mathematical Methods in PhysicsSynopsisMany theories of physics are based on geometrical ideas and concepts. An example is electromagnetism. No sane person would try to write down Maxwell's equations without using div, grad, curl and all that! The indispensable role of vector calculus and Stokes' Theorem in electromagnetism but hints at a deep geometrical structure of the theory.One of the aims of this course is to generalise vector calculus to arbitrary manifolds. This would allow us, for example, to formulate electromagnetism in four-dimensional space-time, so that it is consistent with special relativity. To this end, the concept of a differential manifold, tensors, differential forms and the exterior calculus will be introduced. We then proceed to apply these tools to curved manifolds, a subject known as Riemannian geometry. This forms the basis for the theory of General Relativity, which attributes the gravitational force to the curvature of space-time. Another concept of prime importance in physics is that of symmetry. For example, an object (such as a manifold) may look the same after a translation or rotation: in which case it is said to have translational or rotational symmetry. Symmetries are mathematically described by groups. We shall, in particular, be studying a class of groups known as Lie groups, that have many physical applications. Rotations in space are described by the Lie group SO(3). Other Lie groups, such as U(1), SU(2) and SU(3), play a fundamental role in describing the electromagnetic, weak and strong forces respectively. As this is a course targetted at physicists, we shall not be too concerned with mathematical rigour or formality. Instead, physical applications would be constantly emphasised. At the end of this course, it is hoped that you would not only look at old theories (e.g., electromagnetism, thermodynamics) in a new and revealing light, but be ready to tackle the frontiers of modern physics (e.g., General Relativity, gauge theories, particle physics).
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