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.