BSc Computer Science and Mathematics

Differential and integral calculus, whose origins go back to Newton and Leibniz and their predecessors, has undergone spectacular development in the past centuries to become modern analysis deeply related with multidimensional geometry, topology and abstract algebra. In one aspect, however, it remains close to the founding fathers, namely, in being grounded in applications such as in physics. The course aims to emphasize both abstract and applied sides and to give skills and knowledge for later studies including other year 3 and year 4 courses.

Anti-requisites: MATH20132 Calculus of Several Variables (from 2022-23 or earlier) 

The unit aims to: 

1. Introduce the students to main concepts of modern analysis in multidimensional space such as the idea of differential as the linear operator approximating a function at a given point, and the apparatus of differential forms; 
2. Develop geometric understanding behind the fundamental theorems such as the implicit function theorem and inverse function theorem (local diffeomorphisms, submanifolds specified by equations, curvilinear coordinates, tangent spaces). 

On the successful completion of the course, students will be able to:

• State and use the definitions and main properties of open sets in affine space and of continuous maps. 
• State and use the definition and main properties of the differential of a map, and to work with differentials practically. 
• State and use the inverse function theorem and the implicit function theorem, and to apply them to related geometric concepts.
• Calculate with differential forms including their exterior multiplication and integration. 
• Calculate exterior differentials applying them in particular to   integrals by using variants of Ostrogradsky-Gauss-Stokes formula.

Traditional format (2 lectures and 1 tutorial per week).

The possibility is considered of switching to blended format (1 review class and 1 tutorial per week, together with lecture notes, example sheets and videos to watch).

General feedback is available after the exam is marked.

R.Abraham, J.Marsden, T.Ratiu. Manifolds, tensor analysis and applications. - Springer, 2nd corrected ed. 1988

B.A.Dubrovin, A.T.Fomenko, S.P.Novikov. Modern Geometry - Methods and Applications: Part I. - Springer, 2nd ed. 1992

H.Flanders. Differential forms with applications to physical sciences. - Dover, 2003

W.Rudin. Principles of mathematical analysis. - McGraw Hill, 1976

V.A.Zorich. Mathematical analysis II. - Springer, 2008