# MA259 Multivariable Calculus

**Lecturer: **Mario Micallef and Florian Theil

**Term(s): **1

**Status for Mathematics students: **Core

**Commitment: **30 lectures

**Assessment: **85% by examination, 15% by assignments

**Prerequisites: **MA131 Analysis I and II OR MA137 Mathematical Analysis, MA106 Linear Algebra, MA134 Geometry and Motion OR PX129 Tutorial

**Leads To: ** MA209 Variational Principles, MA3D9 Geometry of Curves and Surfaces, MA3G7 Functional Analysis I, MA3G8 Functional Analysis II, MA3H5 Manifolds, MA3J3 Bifurcations Catastrophes and Symmetry.

**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.