Introductory description

N/A

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• 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 R3
• Maxima, minima and saddles

Learning outcomes

By the end of the module, students should be able to:

• 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.

Indicative reading list

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.
7. J. R. Munkres. Analysis on Manifolds. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1991.
8. W. Rudin. Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-Düsseldorf, third edition, 1976.
9. M. Spivak. Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus. W. A. Benjamin, Inc., New York-Amsterdam, 1965.

Subject specific skills

Students will gain knowledge of definitions, theorems and calculations in 
-Continuous vector-valued functions
-Differentiable functions
-Inverse and Implicit Function Theorems
-Vector fields
-Maxima, minima and saddles

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.