## Vector

A vector is a set of numbers , ..., that transform as

 (1)

This makes a vector a Tensor of Rank 1. Vectors are invariant under Translation, and they reverse sign upon inversion.

A vector is uniquely specified by giving its Divergence and Curl within a region and its normal component over the boundary, a result known as Helmholtz's Theorem (Arfken 1985, p. 79). A vector from a point to a point is denoted , and a vector may be denoted , or more commonly, .

A vector with unit length is called a Unit Vector and is denoted with a Hat. An arbitrary vector may be converted to a Unit Vector by dividing by its Norm, i.e.,

 (2)

Let be the Unit Vector defined by

 (3)

Then the vectors , a, b, c, d satisfy the identities
 (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

and

 (14)

where is the Kronecker Delta, is a Dot Product, and Einstein Summation has been used.

See also Four-Vector, Helmholtz's Theorem, Norm, Pseudovector, Scalar, Tensor, Unit Vector, Vector Field

References

Arfken, G. Vector Analysis.'' Ch. 1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 1-84, 1985.

Aris, R. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. New York: Dover, 1989.

Crowe, M. J. A History of Vector Analysis: The Evolution of the Idea of a Vectorial System. New York: Dover, 1985.

Gibbs, J. W. and Wilson, E. B. Vector Analysis: A Text-Book for the Use of Students of Mathematics and Physics, Founded Upon the Lectures of J. Willard Gibbs. New York: Dover, 1960.

Marsden, J. E. and Tromba, A. J. Vector Calculus, 4th ed. New York: W. H. Freeman, 1996.

Morse, P. M. and Feshbach, H. Vector and Tensor Formalism.'' §1.5 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 44-54, 1953.

Schey, H. M. Div, Grad, Curl, and All That: An Informal Text on Vector Calculus. New York: Norton, 1973.

Schwartz, M.; Green, S.; and Rutledge, W. A. Vector Analysis with Applications to Geometry and Physics. New York: Harper Brothers, 1960.

Spiegel, M. R. Theory and Problems of Vector Analysis. New York: Schaum, 1959.