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Schwere, Elektricität und Magnetismus:395

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Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.


from the nature of the formula itself. The most important discontinuities of scalars are those which occur at surfaces: in the case of vectors, discontinuities at surfaces, at lines, and at points, should be considered.

     74. From equation (3) we obtain



where the accents distinguish the quantities relating to the limits of the line-integrals. We are thus able to reduce a line-integral of the form to the form with quantities free from the sign of integration.

     75. From equation (5) we obtain



where, as elsewhere in these 'equations, the line-integral relates to the boundary of the surface integral.

     From this, by substitution of for , we may derive as a particular case



     76. From equation (4) we obtain



where, as elsewhere in these equations, the surface-integral relates to the boundary of the volume-integrals.

     From this, by substitution of for , we derive as a particular case



which is Green’s Theorem. The substitution of for gives the more general form of this theorem which is due to Thomson, viz:—




     77. From equation (6) we obtain



A particular ease is



Integration of Differential Equations.


     78. If throughout any continuous space (or in all space)