5: Real-Valued Functions of Several Variables

IN THIS CHAPTER we consider real-valued function of \(n\) variables, where \(n>1\).

• SECTION 5.1 deals with the structure of \(\R^ n\), 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 \(\R^n\), including Schwarz’s inequality and the triangle inequality. We define neighborhoods and open sets in \(\R^n\), define convergence of a sequence of points in \(\R^n\), and extend the Heine–Borel theorem to \(\R^n\). The section concludes with a discussion of connected subsets of \(\R^n\).
• 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 \(\R^n\).
• 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.

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