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:

• MA142 Calculus 1
• 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.
• 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.
• MA133 Differential Equations: partial derivatives, multiple integrals, parameterisation of curves and surfaces, arclength and area, line and surface integrals, vector fields.
• 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
• 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.

 

Additional Resources