# Vector Field Theory; the Theorems of Green and Stokes

• Murray H. Protter
• Charles B. MorreyJr.
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Abstract

Let $$\overrightarrow {OP}$$ be the directed line segment in ℝ N having its base at the origin and its head at the point P = (1, 0,..., 0). We define the unit vector e1 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 e1,e2,...,e N . We denote by V N ( 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
$$v = a_1 e_1 + a_2 e_2 + \cdots + a_N e_N$$
where the 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

Scalar Field Vector Function Parametric Representation Piecewise Smooth Divergence Theorem
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