# MA259 Multivariable Calculus

**Lecturer: **Mario Micallef

**Term(s): **Term 1

**Status for Mathematics students: **Core

**Commitment: **30 one-hour lectures plus three 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
- MA448 Hyperbolic Geometry
- MA4J1 Continuum Mechanics
- MA4C0 Differential Geometry
- MA4H0 Applied Dynamical Systems
- MA424 Dynamical Systems
- MA4A2 Advanced Partial Differential 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.