A First Course in Real Analysis pp 404-482 | Cite as

# Vector field theory: The theorems of Green and Stokes

Chapter

## Abstract

Let \(\overrightarrow {OP}\) be the directed line segment in ℝwhere 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 Une 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$$v = {a_1}{e_1} + {a_2}{e_2} + ... + {a_N}{e_N}$$

*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$$\left| v \right| = {{\left( {a_{1}^{2} + a_{2}^{2} + \cdots + a_{N}^{2}} \right)}^{{{{1} \left/ {2} \right.}}}}
$$

## Keywords

Vector Field Vector Function Parametric Representation Piecewise Smooth Simple Closed Curve
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag, New York Inc. 1977