# APPLIED MATHEMATICS PURE MATHEMATICS 4240 DIFFERENTIAL AND INTEGRAL by leg38704

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```									                              APPLIED MATHEM ATICS /PURE MATHEM ATICS 4240
DIFFERENTIAL AND INTEGRAL CALCULUS ON MANIFOLDS
AM/PM 4240: Differential and Integral Calculus on Manifolds
As mathematical and physical problems become more complex, mathematicians rely more and more
heavily on the process of abstraction. This is the process whereby the intrinsic or underlying structures are
revealed and formalized. The process strips away all the superfluous notions that tie the mathematics to
particular realizations or interpretations. Of course, in doing so, the mathematics becomes less easy to
relate to the real world. Calculus on manifolds is an abstraction of calculus that enables us to discuss
relationships and transformations between complicated geometric surfaces and solids and even such
things in higher dimensions (!) in the language of calculus. For example, a function from a sphere to a
cylinder may be said to be differentiable in some precise sense. It is also possible to describe flows and
such things as magnetic or radiation fields around complicated geometries. Much of modern theoretical
physics relies heavily on such a formulation of calculus.

Text. Foundations of Differentiable Manifolds and Lie Groups by Frank Warner.

Marks. The pattern varies with the instructor, but 40% term work and 60% final examination is typical.

Calendar description. Definition and properties of differential manifolds, differentiable maps, tangent
spaces differential of a map, rank of a map, submersion, immersion, submanifolds, Lie group and algebra,
one-parameter subgroups, exponential map, canonical co-ordinates, adjoint representation, Lie
transformation groups, homogeneous spaces of Lie groups, fibre bundles.
Prerequisite: AM/PM 4230.

Offered. Occasionally

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7

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