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Mathematics LibreTexts

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).