MMath Mathematics and Statistics

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.

MATH31061 Requisites: (A) Pre-Requisite - MATH21120; (B) Anti-Requisite - Students who took MATH20132 Calculus of Several Variables are not permitted to take MATH31061 Analysis and Geometry

Students who took MATH20132 Calculus of Several Variables are not permitted to take MATH31061 Analysis and Geometry in Affine Space.

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.

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. Limits, smallness and continuity.

Affine space: points and vectors; R^ as affine space. Affine maps and affine coordinates. Norms of vectors. Equivalence of norms. Limits. Big O and small o. Open sets. Continuous functions: continuity at a point and characterization in terms of open sets.

Differential calculus.

Differential at a point (as a linear map). Coordinate description. Derivative along a vector and partial derivatives. Continuous partial derivatives imply differentiability in a domain. Differentiation of composition. Leibniz rule. Matrix exponential; differentiation of determinant. Finite increment formula. Higher differentials and higher partial derivatives. Hadamard’s lemma and Taylor formula. Critical points.

Implicit function theorem and related topics.

Inverse function theorem. Local diffeomorphism. Local coordinates on an open set of affine space. Implicit function theorem. Submanifolds specified by equations. Local coordinates on them. Tangent spaces.

Integration revisited.

Single-variable integration as integration of a 1-form. Integral over a bounded in R^. Change of variables formula. Volume of a parallelepiped in affine space.

Differential forms.

Rules of wedge product.  Integral of an n-form over an n-dimensional domain. Integral of a k-form over a k-path. Pullback of forms. Forms as multilinear functions of vectors.

Exterior differential.

Variation of integral a path. Definition and fundamental properties of exterior differential.

Stokes formula and its applications.

Orientation and boundary, Ostrogradsky-Gauss theorem for unit cube. Integration over chains and Stokes formula for chains. Classical integral theorems. Poincaré lemma.

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).

 

Formative assessment consisting of marked homework 

General feedback is available after the exam is marked.

Mathematical analysis I    Zorich, V. A.    Springer    2016    ISBN: 9783662487907

Mathematical analysis II    Zorich, V. A.    Springer    2016    ISBN: 9783662489932

Manifolds, Tensor Analysis, and Applications    Abraham, Ralph.    Springer New York    1988    ISBN: 9781461210290

Principles of mathematical analysis    Rudin, Walter,    McGraw-Hill    1976    ISBN: 007054235

Calculus on manifolds : a modern approach to classical theorems of advanced calculus    Spivak, Michael.    Benjamin    1965    
 

Modern geometry : methods and applications    Dubrovin, B. A.    Springer-Verlag    1984    ISBN: 3540908722

Differential Forms with Applications to the Physical Sciences    Harley Flanders    Elsevier Science & Technology    1963    ISBN: 9780122596506