Schwere, Elektricität und Magnetismus:383

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

 With this understanding we may say:—

 The coefficients by which any vector is expressed in terms of three other vectors are the direct products of that vector with the reciprocals of the three.

 Among other relations which are satisfied by reciprocal systems of vectors are the following:

$\alpha \cdot \alpha '=\beta \cdot \beta '=\gamma \cdot \gamma '=1.\,$

(5)

$(\alpha \cdot \beta \times \gamma )(\alpha '\cdot \beta '\times \gamma ')=1.\,$

(6)

(See No. 34.)

$\alpha \times \alpha '+\beta \times \beta '+\gamma \times \gamma '=0.\,$

(7)

(See No. 29.)

 A system of three mutually perpendicular unit vectors is reciprocal to itself, and only such a system.

 The identical equation

$\rho =(\rho \cdot i)i+(\rho \cdot j)j+(\rho \cdot k)k\,$

(8)

may be regarded as a particular case of equation (2).

 The system reciprocal to $\alpha \times \beta ,\,\beta \times \gamma ,\,\gamma \times \alpha$ is

${\frac {\alpha }{\alpha \cdot \beta \times \gamma }},\quad {\frac {\beta }{\alpha \cdot \beta \times \gamma }},\quad {\frac {\gamma }{\alpha \cdot \beta \times \gamma }}.$

 39. Scalar equations of the first degree with respect to an unknown vector.—It is easily shown that any scalar equation of the first degree with respect to an unknown vector $\rho \,$, in which all the other quantities are known, may be reduced to the form

$\rho \cdot \alpha =a,\,$

(8)

in which $\alpha \,$ and $a\,$ are known. (See No. 35.) Three such equations will afford the value of $\rho \,$ (by equation (8) of No. 37, or equation (3) of No. 38), which may be used to eliminate $\rho \,$ from any other equation either scalar or vector.

 When we have four scalar equations of the first degree with respect to $\rho \,$, the elimination may be performed most symmetrically by substituting the values of $\rho \cdot \alpha \,$ etc., in the equation,

$(\rho \cdot \alpha )(\beta \cdot \gamma \times \delta )-(\rho \cdot \beta )(\gamma \cdot \delta \times \alpha )+(\rho \cdot \gamma )(\delta \cdot \alpha \times \beta )-(\rho \cdot \delta )(\alpha \cdot \beta \times \gamma ),\,$

which is obtained from equation (8) of No. 37 by multiplying directly by $\delta \,$. It may also be obtained from equation (5) of No. 37 by writing $\delta \,$ for $\rho \,$, and then multiplying directly by $\rho \,$.

 40. Solution of a vector equation of the first degree with respect to the unknown vector.—It is now easy to solve an equation of the form

$\delta =\alpha (\lambda \cdot \rho )+\beta (\mu \cdot \rho )+\gamma (\nu \cdot \rho ),\,$

(1)