A First Course in Real Analysis pp 413-493 | Cite as

# Vector Field Theory; the Theorems of Green and Stokes

Chapter

## Abstract

Let \(
\overrightarrow {OP}
\) be the directed line segment in ℝ where the

^{ N }having its base at the origin and its head at the point*P*= (1, 0,..., 0). We define the unit vector*e*_{1}as the equivalence class of all directed line segments of length 1 which are parallel to \( \overrightarrow {OP} \) and directed similarly. By considering directed line segments from the origin to the points (0, 1, 0,..., 0), (0, 0, 1,..., 0),..., (0, 0,..., 0, 1), we obtain the set of unit vectors*e*_{1},*e*_{2},...,*e*_{ N }. We denote by*V*_{ N }(*ℝ*^{ N }) or simply*V*_{ N }the linear space formed by taking all linear combinations of these unit vectors with real scalars. That is, any vector*v*in*V*_{ N }is of the form$$
v = a_1 e_1 + a_2 e_2 + \cdots + a_N e_N
$$

*a*_{ i }are real numbers. Addition of vectors and multiplication of vectors by scalars follow the usual rules for a linear space and are a direct generalization of the rules for vectors in two and three dimensions which the reader has encountered earlier.^{1}The**length**of a vector, denoted |*v*|, is$$
\left| v \right| = \left( {a_1^2 + a_2^2 + \cdots + a_N^2 } \right)^{1/2} .
$$

### Keywords

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## Copyright information

© Springer Science+Business Media New York 1991