5: Real-Valued Functions of Several Variables
( \newcommand{\kernel}{\mathrm{null}\,}\)
IN THIS CHAPTER we consider real-valued function of n variables, where n>1.
- SECTION 5.1 deals with the structure of \Rn, the space of ordered n-tuples of real numbers, which we call {}. We define the sum of two vectors, the product of a vector and a real number, the length of a vector, and the inner product of two vectors. We study the arithmetic properties of \Rn, including Schwarz’s inequality and the triangle inequality. We define neighborhoods and open sets in \Rn, define convergence of a sequence of points in \Rn, and extend the Heine–Borel theorem to \Rn. The section concludes with a discussion of connected subsets of \Rn.
- SECTION 5.2 deals with boundedness, limits, continuity, and uniform continuity of a function of n variables; that is, a function defined on a subset of \Rn.
- SECTION 5.3 defines directional and partial derivatives of a real-valued function of n variables. This is followed by the definition of differentiablity of such functions. We define the differential of such a function and give a geometric interpretation of differentiablity.
- SECTION 5.4 deals with the chain rule and Taylor’s theorem for a real-valued function of n variables.