Skip to main content
$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 1: Vectors in Euclidean Space

In vector (or multivariable) calculus, we will deal with functions of two or three variables (usually $$x, y$$ or $$x, y, z$$, respectively). The graph of a function of two variables, say, $$z = f(x,y)$$, lies in Euclidean space, which in the Cartesian coordinate system consists of all ordered triples of real numbers $$(a, b, c)$$. Since Euclidean space is 3-dimensional, we denote it by $$\mathbb{R}^{3}$$. The graph of $$f$$ consists of the points $$(x, y, z) = (x, y, f(x, y))$$.

Thumbnail: Illustration of the Cartesian coordinate system for 3D. Image used with permission (Public Domain; Jorge Stolfi).