Zum Inhalt springen

Schwere, Elektricität und Magnetismus:404

aus Wikisource, der freien Quellensammlung
Bernhard Riemann: Schwere, Elektricität und Magnetismus
Seite 32
<< Zurück Vorwärts >>
fertig
Fertig! Dieser Text wurde zweimal anhand der Quelle Korrektur gelesen. Die Schreibweise folgt dem Originaltext.


VECTOR ANALYSIS.


space. It might seem harmless to set an indefinite expression equal to a definite, but it would be dangerous, since we might with equal right set the indefinite expression equal to other definite expressions, and then be misled into supposing these definite expressions to be equal to one another. It will be safe to say that the above equations will hold, provided that the potential of or has a definite value. It will be observed that whenever or has a definite value in general, (i. e. with the possible exception of certain points, lines, and surfaces),*[1] the first members of all these equations will have definite values in general, and therefore the second members of the equations, being necessarily equal to the first members, when these have definite values, will also have definite values in general.

     94. Again, whenever has a definite value, we may write



where stands for . But



whence



     Moreover, will in general have a definite value, if has.

     95. In like manner, whenever Pot has a definite value,



Substituting the value of given above we have



      will have a definite value in general, whenever has.

     96. Hence, with the aid of No. 93, we obtain




whenever has a definite value.

     97. By. the method of No. 93 we obtain



  1. * Whenever it is said that a function of position in space has a definite value in general, this phrase is to be understood as explained above. The term definite is intended to exclude both indeterminate and infinite values.