Schwere, Elektricität und Magnetismus:392

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

on which a positive motion in the circle appears counter-clockwise, gives the direction of $\nabla \times \omega \,$, and the quotient of the integral divided by the area of the circle gives the magnitude of $\nabla \times \omega \,$.

$\nabla \,$, $\nabla \cdot \,$, and $\nabla \times \,$ applied to Functions of Functions of Position.

     62. A constant scalar factor after $\nabla \,$, $\nabla \cdot \,$ or $\nabla \times \,$ may he placed before the symbol.

     63. If $f(u)\,$ denotes any scalar function of $u\,$, and $f'(u)\,$ the derived function,

$\nabla f(u)=f'(u)\nabla u.$

     64. If $u\,$ or $\omega \,$ is a function of several scalar or vector variables, which are themselves functions of the position of a single point, the value of $\nabla u\,$ or $\nabla \cdot \omega \,$ or $\nabla \times \omega \,$ will be equal to the sum of the values obtained by making successively all but each one of these variables constant.

     65. By the use of this principle, we easily derive the following identical equations:

$\nabla (t+u)=\nabla t+\nabla u.$

(1)

$\nabla \cdot (\tau +\omega )=\nabla \cdot \tau +\nabla \cdot \omega .\quad \nabla \times \lbrack \tau +\omega \rbrack =\nabla \times \tau +\nabla \times \omega .\,$

(2)

$\nabla (tu)=u\nabla t+t\nabla u.\,$

(3)

$\nabla \cdot (u\omega )=\omega \cdot \nabla u+u\cdot \nabla \omega .\,$

(4)

$\nabla \times \lbrack u\omega \rbrack =u\nabla \times \omega -\omega \times \nabla u.\,$

(5)

$\nabla \cdot \lbrack \tau \times \omega \rbrack =\omega \cdot \nabla \times \tau -\tau \cdot \nabla \times \omega .\,$

(6)

     The student will observe an analogy between these equations and the formulae of multiplication. (In the last four equations the analogy appears most distinctly when we regard all the factors but one as constant.) Some of the more curious features of this analogy are due to the fact that the $\nabla \,$ contains implicitly the vectors $i,\,j\,$ and $k\,$, which are to be multiplied into the following quantities.

Combinations of the Operators $\nabla \,$, $\nabla \cdot \,$ and $\nabla \times \,$.

     66. If $u\,$ is any scalar function of position in space,

$\nabla \times \nabla u=0,\,$

as may be derived directly from the definitions of these operators.

     67. Conversely, if $\omega \,$ is such a vector function of position in space that

$\nabla \times \omega =0,\,$