Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/66

50 Whenever, therefore, $$\omega$$ is discontinuous at surfaces, the expressions $$\text{Pot }\nabla. \omega$$ and $$\text{New } \nabla. \omega$$ must be regarded as implicitly including the surface-integrals respectively, relating to such surfaces, and the expressions $$\text{Pot } \nabla \times \omega$$ and $$\text{Lap } \nabla \times \omega$$ as including the surface-integrals  respectively, relating to such surfaces.

101. We have already seen that if $$\omega$$ is the curl of any vector function of position, $$\nabla. \omega = 0.$$ (No. 68.) The converse is evidently true, whenever the equation $$\nabla. \omega = 0$$ holds throughout all space, and $$\omega$$ has in general a definite potential; for then Again, if $$\nabla. \omega = 0$$ within any aperiphractic space $$\text{A},$$ contained within finite boundaries, we may suppose that space to be enclosed by a shell $$\text{B}$$ having its inner surface coincident with the surface of $$\text{A}.$$ We may imagine a function of position $$\omega ',$$ such that $$\omega ' = \omega$$ in $$\text{A}, \omega ' = 0$$ outside of the shell $$\text{B}'$$ and the integral $$\iiint \omega '. \omega ' dv$$ for $$\text{B}$$ has the least value consistent with the conditions that the normal component of $$\omega '$$ at the outer surface is zero, and at the inner surface is equal to that of $$\omega,$$ and that in the shell $$\nabla. \omega ' = 0$$ (compare No. 90). Then $$\nabla. \omega ' = 0$$ throughout all space, and the potential of $$\omega '$$ will have in general a definite value. Hence, and $$\omega$$ will have the same value within the space $$\text{A}.$$

102. Def.—If $$\omega$$ is a vector function of position in space, the Maxwellian of $$\omega$$ is a scalar function of position, defined by the equation (Compare No. 92.) From this definition the following properties are easily derived. It is supposed that the functions $$\omega$$ and $$u$$ are such that their potentials have in general definite values.