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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:

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


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


  • 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


  • 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