Analysis and Geometry in Affine Space

MATH31061 Analysis and Geometry in Affine Space 10 Semester 1 Undergraduate Level 3 Theodore Voronov Unit coordinator 1 FHEQ level (Framework for Higher Education Qualifications) ' Last part of a Bachelors ' European Credit Transfer & Accumulation System Rating : 5.0 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. 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.  Syllabus:Affine space and continuous functions.  Affine space: points and vectors; R^n as affine space; affine coordinates. Open sets: norms of vectors; equivalence of different norms; definition of an open set; properties of open sets. Limits. Big O and little o. Continuous functions: continuity at a point and characterization in terms of open sets. Continuity of composition. Algebra of continuous functions.Differentiation.Differentiability at a point. Differential as a linear map. Derivative along a vector and partial derivatives; coordinate description of differential. Continuous partial derivatives imply differentiability in a domain. Differentiation of composition. Product rule for functions with values in an algebra. Examples: matrix exponential; differentiation of determinant. Higher differentials and derivatives; functions of classes C^k. Taylor formula. Critical points.Implicit function theorem and its applications.Inverse function theorem and local diffeomorphism. Implicit function theorem. Submanifolds specified by equations; local coordinates. Curvilinear coordinates on an open set of affine space. Tangent spaces. “Moving” bases induced by local coordinates on a submanifold or by curvilinear coordinates.Integration and differential forms.Revisiting single-variable integration as integration of 1-forms. Volume of a parallelepiped in affine space. Integral of a function over a bounded domain with respect to an affine measure. Change of variables formula. Orientation. Rules of wedge product.  Integral of an n-form over an n-dimensional domain. Integral of a k-form over a parametrized k-submanifold. Forms as multilinear functions of vectors. Pullback of forms.Stokes formula and exterior differential.Motivating example: integral over the boundary of a small rectangle. Definition and properties of exterior differential. Stokes(-Ostrogradsky-Gauss) formula for a cube. Integration over chains and Stokes formula for chains. Variants of Stokes formula (e.g. for domains with good descriptions of boundary). Further applications of exterior differential: e.g. Poincaré lemma, relation with commutator of vector fields, Cartan's formulas, Frobenius theorem.  The unit aims to: <ol><li>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; </li><li>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). </li></ol> On the successful completion of the course, students will be able to:<ul><li>State and use the definitions and main properties of open sets in affine space and of continuous maps. </li><li>State and use the definition and main properties of the differential of a map, and to work with differentials practically. </li><li>State and use the inverse function theorem and the implicit function theorem, and to apply them to related geometric concepts.</li><li>Calculate with differential forms including their exterior multiplication and integration. </li><li>Calculate exterior differentials applying them in particular to   integrals by using variants of Ostrogradsky-Gauss-Stokes formula.</li></ul> 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).  1 Written exam 100% Formative assessment consisting of marked homework  General feedback is available after the exam is marked. MATH20132 Calculus of Several Variables Anti-requisite Compulsory MATH21120 Groups and Geometry Pre-Requisite Compulsory MATH20111 Real Analysis B Pre-Requisite Compulsory MATH20101 Real Analysis A Pre-Requisite Compulsory MATH31061 Requisites: (A) Pre-Requisite - MATH21120 or MATH20101/11. (B) Anti-Requisite - Students who took MATH20132 Calculus of Several Variables are not permitted to take MATH31061 Analysis and GeometryAnti-requisites: MATH20132 Calculus of Several Variables (from 2022-23 or earlier)  N 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  Lectures 22 Tutorials 11 0 67