Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/33

III.

ELEMENTS OF VECTOR ANALYSIS.

[Privately printed, New Haven, pp. 17–50, 1881; pp. 50–90, 1884.]

(The fundamental principles of the following analysis are such as are familiar under a slightly different form to students of quaternions. The manner in which the subject is developed is somewhat different from that followed in treatises on quaternions, since the object of the writer does not require any use of the conception of the quaternion, being simply to give a suitable notation for those relations between vectors, or between vectors and scalars, which seem most important, and which lend themselves most readily to analytical transformations, and to explain some of these transformations. As a precedent for such a departure from quaternionic usage, Clifford's Kinematic may be cited. In this connection, the name of Grassmann may also be mentioned, to whose system the following method attaches itself in some respects more closely than to that of Hamilton.)

CHAPTER I.

concerning the algebra of vectors.

Fundamental Notions.

1. Definition.—If anything has magnitude and direction, its magnitude and direction taken together constitute what is called a vector.

The numerical description of a vector requires three numbers, but nothing prevents us from using a single letter for its symbolical designation. An algebra or analytical method in which a single letter or other expression is used to specify a vector may be called a vector algebra or vector analysis.

Def.—As distinguished from vectors the real (positive or negative) quantities of ordinary algebra are called scalars.^[1]

As it is convenient that the form of the letter should indicate whether a vector or a scalar is denoted, we shall use the small Greek letters to denote vectors, and the small English letters to denote scalars. (The three letters, $i,j,k$, will make an exception, to be mentioned more particularly hereafter. Moreover, $\pi$ will be used in its usual scalar sense, to denote the ratio of the circumference of a circle to its diameter.)

1. The imaginaries of ordinary algebra may be called biscalars, and that which corresponds to them in the theory of vectors, bivectors. But we shall have no occasion to consider either of these. [See, however, footnote on p. 84.]