Euclidean Space

Euclidean $n$-space is the Space of all $n$-tuples of Real Numbers, ($x_1$, $x_2$, ..., $x_n$) and is denoted $\Bbb{R}^n$. $\Bbb{R}^n$ is a Vector Space and has Lebesgue Covering Dimension $n$. Elements of $\Bbb{R}^n$ are called $n$-Vectors. $\Bbb{R}^1=\Bbb{R}$ is the set of Real Numbers (i.e., the Real Line), and $\Bbb{R}^2$ is called the Euclidean Plane. In Euclidean space, Covariant and Contravariant quantities are equivalent so $\vec e^{\,j} = \vec e_j$.

See also Euclidean Plane, Real Line, Vector

References

Gray, A. ``Euclidean Spaces.'' §1.1 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 2-4, 1993.