Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/64

49 99. To assist the memory of the student, some of the principal results of Nos. 93–98 may be expressed as follows:

Let $$\omega_{1}$$ be any solenoidal vector function of position in space, $$\omega_{2}$$ any irrotational vector function, and $$u$$ any scalar function, satisfying the conditions that their potentials have in general definite values.

With respect to the solenoidal function $$\omega_{1}, \frac{1}{4\pi} \text{Lap}$$ and $$\nabla \times$$ are inverse operators; i.e., Applied to the irrotational function $$\omega_{2},$$ either of these operators gives zero; i.e.,  With respect to the irrotational function $$\omega_{2},$$ or the scalar function $$u, \frac{1}{4\pi} \text{New}$$ and $$- \nabla .$$ are inverse operators; i.e.,  Applied to the solenoidal function $$\omega_{1},$$ the operator $$\nabla .$$ gives zero; i.e.  Since the most general form of a vector function having in general a definite potential may be written $$\omega_{1} + \omega_{2},$$ the effect of these operators on such a function needs no especial mention.

With respect to the solenoidal function $$\omega_{1}, \frac{1}{4\pi} \text{Pot }$$ and $$\nabla \times \nabla \times$$ are inverse operators; i.e., With respect to the irrotational function $$\omega_{2}, \frac{1}{4\pi} \text{Pot }$$ and $$- \nabla \nabla .$$ are inverse operators; i.e.,  With respect to any scalar or vector function having in general a definite potential $$\frac{1}{4\pi} \text{Pot}$$ and $$- \nabla. \nabla$$ are inverse operators; i.e., With respect to the solenoidal function $$\omega_{1}, -\nabla. \nabla$$ and $$\nabla \times \nabla \times$$ are equivalent; with respect to the irrotational function $$\omega_{2}, \nabla. \nabla$$ and $$\nabla \nabla .$$ are equivalent; i.e.,