Lecturer: Mario Micallef 

Term(s): Term 1

Status for Mathematics students: Core

Commitment: 30 one-hour lectures plus assignments

Assessment: 85% 2-hour examination, 15% coursework

Formal registration prerequisites: None

Assumed knowledge:

• MA131 Analysis I & II or MA137 Mathematical Analysis: epsilon-delta definition of continuity and continuous limits, properties of continuous functions, definition of derivative, Mean Value Theorem, Taylor's theorem with remainder, supremum and infimum.
• MA106 Linear Algebra: Rank-Nullity Theorem and its geometric interpretation, dependence of matrix representation of a linear map with respect to a choice of bases, determinant.
• MA134 Geometry and Motion: partial derivatives, multiple integrals, parameterisation of curves and surfaces, arclength and area, line and surface integrals, vector fields.

A much more detailed list of prerequisites will be posted on the module's webpage a couple of weeks before the beginning of term.

Useful knowledge: Plotting graphs and contour plots of simple functions of two variables; the use of appropriate mathematical software for this purpose is encouraged.

Synergies:

• MA244 Analysis III - particularly the Complex Analysis section
• MA251 Algebra I: Advanced Linear Algebra - particularly bilinear forms and orthogonal matrices
• MA250 Introduction to Partial Differential Equations
• MA260 Norms, Metrics and Topologies or MA222 Metric Spaces
• MA209 Variational Principles
• MA3D9 Geometry of Curves and Surfaces
• MA3H5 Manifolds as well as all PDEs and fluids modules

Leads to: The following modules have this module listed as assumed knowledge or useful background:

• MA222 Metric Spaces
• MA260 Norms, Metrics and Topologies
• MA250 Introduction to Partial Differential Equations
• MA254 Theory of ODEs
• MA261 Differential Equations: Modelling and Numerics
• MA269 Asymptotics and Integral Transforms
• MA209 Variational Principles
• MA3H0 Numerical Analysis and PDEs
• MA3J3 Bifurcations, Catastrophes and Symmetry
• MA3D9 Geometry of Curves and Surfaces
• MA3G8 Functional Analysis II
• MA3K0 High Dimensional Probability
• MA398 Matrix Analysis and Algorithms
• MA3H5 Manifolds
• MA3K1 Mathematics of Machine Learning
• MA3D1 Fluid Dynamics
• MA3B8 Complex Analysis
• MA3G1 Theory of Partial Differential Equations
• MA3H7 Control Theory
• MA3G7 Functional Analysis I
• MA448 Hyperbolic Geometry
• MA4J1 Continuum Mechanics
• MA4C0 Differential Geometry
• MA4H0 Applied Dynamical Systems
• MA424 Dynamical Systems
• MA4A2 Advanced Partial Differential Equations
• MA4L9 Variational Analysis and Evolution Equations

Content:

• Continuous Vector-Valued Functions
• Some Linear Algebra
• Differentiable Functions
• Inverse Function Theorem and Implicit Function Theorem
• Vector Fields, Green’s Theorem in the Plane and the Divergence Theorem in $\mathbb{R}^3$
• Maxima, minima and saddles

Learning Outcomes:

• Demonstrate understanding of the basic concepts, theorems and calculations of multivariate analysis
• Demonstrate understanding of the Implicit and Inverse Function Theorems and their applications
• Demonstrate understanding of vector fields and Green’s Theorem and the Divergence Theorem
• Demonstrate the ability to analyse and classify critical points using Taylor expansions

Books:

1. R. Abraham, J. E. Marsden, T. Ratiu. Manifolds, Tensor Analysis, and Applications. Springer, second edition, 1988.
2. T. M. Apostol. Mathematical Analysis. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., second edition, 1974.
3. R. Coleman. Calculus on Normed Vector Spaces, Springer 2012. [available online via Warwick's library]
4. J. J. Duistermaat, J. A. C. Kolk. Multidimensional Real Analysis I : Differentiation, CUP, 2004 [available online via Warwick's library].
5. T. W. Körner. A Companion to Analysis: A Second First and First Second Course in Analysis, volume 62 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2004.
6. J. E. Marsden and A. Tromba. Vector Calculus. Macmillan Higher Education, sixth edition, 2011.

Additional Resources