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.