5: Real-Valued Functions of Several Variables
- Page ID
- 33457
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
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.