# MA263 Multivariable Analysis

**Lecturer: **Felix Schulze

**Term(s):** Term 2

**Status for Mathematics students:** Core for MMath G103, Optional Core for BSc G100

**Commitment:** 30 one-hour lectures plus assignments

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

**Formal registration prerequisites: **None

**Assumed knowledge: **

- MA141 Analysis 1 or MA140 Mathematical Analysis 1 or MA142 Calculus 1
- MA139 Analysis 2 or MA152 Mathematical Analysis 2 or MA143 Calculus 2: 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.
- MA150 Algebra 2 or MA149 Linear Algebra or MA148 Vectors and Matrices: Rank-Nullity Theorem and its geometric interpretation, dependence of matrix representation of a linear map with respect to a choice of bases, determinant.
- MA144 Methods of Mathematical Modelling 2 or MA145 Mathematical Methods and Modelling 2 or MA133 Differential Equations: partial derivatives, multiple integrals, parameterisation of curves and surfaces, arclength and area, line and surface integrals, vector fields.
- MA270 Analysis 3 or MA271 Mathematical Analysis 3 : Differentiable Functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ and uniform convergence.

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

**Synergies:**

- MA266 Multilinear Algebra - particularly bilinear forms and orthogonal matrices
- MA250 Introduction to Partial Differential Equations
- MA260 Norms, Metrics and Topologies or MA222 Metric Spaces
- MA269 Asymptotics and Integral Transforms

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

- 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:**

- Differentiable Functions from $\mathbb{R}^n$ to $\mathbb{R}^m$
- Inverse Function Theorem and Implicit Function Theorem
- Higher Dimensinal Riemann Integration
- Vector Fields, Green’s Theorem in the plane, Stokes' Theorem on 2-dimensional surfaces and the Divergence Theorem in $\mathbb{R}^3$
- Taylor’s theorem in higher dimensions and 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: **

- R. Abraham, J. E. Marsden, T. Ratiu. Manifolds,
*Tensor Analysis, and Applications*. Springer, second edition, 1988. - T. M. Apostol.
*Mathematical Analysis*. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., second edition, 1974. - R. Coleman.
*Calculus on Normed Vector Spaces*, Springer 2012. [available online via Warwick's library] - J. J. Duistermaat, J. A. C. Kolk.
*Multidimensional Real Analysis I : Differentiation*, CUP, 2004 [available online via Warwick's library]. - 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. - J. E. Marsden and A. Tromba.
*Vector Calculus*. Macmillan Higher Education, sixth edition, 2011.