Functions of several variables

1. Differentiable Functions.- 1.1 Introduction.- 1.2 Linear part of a function.- 1.3 Vector viewpoint.- 1.4 Directional derivative.- 1.5 Tangent plane to a surface.- 1.6 Vector functions.- 1.7 Functions of functions.- 2. Chain Rule and Inverse Function Theorem.- 2.1 Norms.- 2.2 Fréchet derivatives.- 2.3 Chain rule.- 2.4 Inverse function theorem.- 2.5 Implicit functions.- 2.6 Functional dependence.- 2.7 Higher derivatives.- 3. Maxima and Minima.- 3.1 Extrema and stationary points.- 3.2 Constrained minima and Lagrange multipliers.- 3.3 Discriminating constrained stationary points.- 3.4 Inequality constraints.- 3.5 Discriminating maxima and minima with inequality constraints 62 Further reading.- 4. Integrating Functions of Several Variables.- 4.1 Basic ideas of integration.- 4.2 Double integrals.- 4.3 Length, area and volume.- 4.4 Integrals over curves and surfaces.- 4.5 Differential forms.- 4.6 Stokes's theorem.- Further reading.- Appendices.- A. Background required in linear algebra and elementary calculus.- B. Compact sets, continuous functions and partitions of unity.- C. Answers to selected exercises.- Index (including table of some special symbols).