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Schwere, Elektricität und Magnetismus:400

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Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.


     Setting , we have throughout the space, and the normal component of at the boundary equal to zero. Hence throughout the whole space .

     89. If throughout a certain space (which need not be continuous, and which may extend to infinity)



and in all the bounding surfaces



and at infinite distances within the space (if such there are)



then throughout the whole space



This will be apparent if we consider separately each of the scalar components of and .


Minimum Values of the Volume-integral .
(Thomson’s Theorems.)


     90. Let it he required to determine for a certain space a vector function of position subject to certain conditions (to be specified hereafter), so that the volume-integral



for that space shall have a minimum value, denoting a given positive scalar function of position.

     a. In the first place, let the vector be subject to the conditions that is given within the space, and that the normal component of is given for the bounding surface. (This component must of course be such that the surface-integral of shall be equal to the volume-integral . If the space is not continuous, this must be true of each continuous portion of it. See No. 57.) The solution is that , or more generally, that the line-integral of for any closed curve in the space shall vanish.

     The existence of the minimum requires that



while is subject to the limitation that



and that the normal component of at the bounding surface vanishes. To prove that the line-integral of vanishes for