Vector

A vector is a set of numbers $A_0$, ..., $A_n$ that transform as

$$A_i' = a_{ij}A_j.$$ (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 $A$ to a point $B$ is denoted $\overrightarrow{AB}$, and a vector $v$ may be denoted $\vec v$, or more commonly, ${\bf v}$.

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.,

$$\hat{\bf v}\equiv {{\bf v}\over \vert{\bf v}\vert}.$$ (2)

Let $\hat{\bf n}$ be the Unit Vector defined by

$$\hat{\bf n}\equiv\left[{\matrix{\cos \theta \sin \phi \cr \sin \theta \sin \phi \cr \cos \phi \cr}}\right].$$ (3)

Then the vectors $\hat{\bf n}$, a, b, c, d satisfy the identities

$$\left\langle{n_x}\right\rangle{} = \int_0^{2\pi}\int_0^\pi (\cos \theta \sin \phi )\sin \phi \,d\theta \,d\phi$$

$$= [\sin \theta]^{2\pi}_0 \int_0^{2\pi} \sin^2\phi \,d\phi =0$$ (4)

$$\left\langle{n_i}\right\rangle{} = 0$$ (5)

$$\left\langle{n_in_j}\right\rangle{} = {\textstyle{1\over 3}}\delta_{ij}$$ (6)

$$\left\langle{n_in_kn_k}\right\rangle{} = 0$$ (7)

$$\left\langle{n_in_kn_ln_m}\right\rangle{} = {\textstyle{1\over 15}}(\delta_{ik}\delta_{lm}+\delta_{il}\delta_{km}+\delta_{im}\delta_{kl})$$ (8)

$$\left\langle{({\bf a}\cdot \hat{\bf n})^2}\right\rangle{} = {\textstyle{1\over 3}}a^2$$ (9)

$$\left\langle{({\bf a}\cdot \hat{\bf n})({\bf b}\cdot \hat{\bf n})}\right\rangle{} = {\textstyle{1\over 3}}{\bf a}\cdot{\bf b}$$ (10)

$$\left\langle{({\bf a}\cdot \hat{\bf n})\hat{\bf n}}\right\rangle{} = {\textstyle{1\over 3}}a$$ (11)

$$\left\langle{({\bf a}\times\hat{\bf n})^2}\right\rangle{} = {\textstyle{2\over 3}}a^2$$ (12)

$$\left\langle{({\bf a}\times\hat{\bf n})\cdot({\bf b}\times\hat{\bf n})}\right\rangle{} = {\textstyle{2\over 3}}{\bf a}\cdot{\bf b},$$ (13)

and

$$\left\langle{({\bf a}\cdot{\hat{\bf n}})({\bf b}\cdot{\hat{\... ...bf b}\cdot{\bf d})+({\bf a}\cdot{\bf d})({\bf b}\cdot{\bf c})]$$ (14)

where $\delta_{ij}$ is the Kronecker Delta, ${\bf a}\cdot{\bf b}$ 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.