# MA263 Content

**Content:**

- 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

**Books: **

- 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.