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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:

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

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