Schwere, Elektricität und Magnetismus:390

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

This quotient must therefore be independent of the form of the surface. We may define $\nabla \cdot \omega \,$ as representing that quotient, and then obtain equation (1) of No. 54 by applying the general principle to the ease of the rectangular parallelopiped.

     56. Skew surface-integrals.—The integral $\iint d\sigma \times \omega \,$ may be called the skew surface-integral of $\omega \,$. It is evidently a vector. For a closed surface bounding a space $dv\,$ infinitely small in all dimensions, this integral reduces to $\nabla \times \omega dv\,$, as is easily shown by reasoning like that of No. 55.

     57. Integration.—If $dv\,$ represents an element of any space, and $d\sigma \,$ an element of the bounding surface,

$\iiint \nabla \cdot \omega dv=\iint \omega \cdot d\sigma .\,$

For the first member of this equation represents the sum of the surface integrals of all the elements of the given space. We may regard this principle as affording a means of integration, since we may use it to reduce a triple integral (of a certain form) to a double integral.

     The principle may also be expressed as follows:

     The surface-integral of any vector function of position in space for a closed surface is equal to the volume-integral of the divergence of that function for the space enclosed.

     58. Line-integrals.—The integral $\int \omega \cdot d\rho \,$, in which $d\rho \,$ denotes the element of a line, is called the line-integral of $\omega \,$ for that line. It is implied that one of the directions of the line is distinguished as positive. When the line is regarded as bounding a surface, that side of the surface will always be regarded as positive, on which the surface appears to be circumscribed counter-clock-wise.

     59. Integration.—From No. 51 we obtain directly

$\int \nabla \cdot ud\rho =u''-u',\,$

where the single and double accents distinguish the values relating to the beginning and end of the line.

     In other words,—The line-integral of the derivative of any (continuous) scalar function of position in space is equal to the difference of the values of the function at the extremities of, the line. For a closed line the integral vanishes.

     60. Integration.—The following principle may be used to reduce double integrals of a certain form to simple integrals.

     If $d\sigma \,$ represents an element of any surface, and $d\rho \,$ an clement of the bounding line,

$\iint \nabla \times \omega \cdot d\sigma =\int \omega \cdot d\rho .\,$

In other words,—The line-integral of any vector function of position in space for a closed line is equal to the surface-inte-