“Vector”的概念、定义、习题解题方法及翻译-数学辞典

单词

Vector

释义

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 componentover 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 beconverted 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

Vectors

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.

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