Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/91

Rh

169. If $$\omega$$ is a vector having continuously varying values in space, and $$\rho$$ the vector determining the position of a point, we may set and regard $$\omega$$ as a function of $$\rho$$ or of $$x, y,$$ and $$z.$$ Then,  that is,  If we set  Here $$\nabla$$ stands for  exactly as in No. 52, except that it is here applied to a vector and produces a dyadic, while in the former case it was applied to a scalar and produced a vector. The dyadic $$\nabla \omega$$ represents the nine differential coefficients of the three components of w with respect to $$x, y,$$ and $$z,$$ just as the vector $$\nabla u$$ (where $$u$$ is a scalar function of $$\rho$$) represents the three differential coefficients of the scalar $$u$$ with respect to $$x, y,$$ and $$z.$$

It is evident that the expressions $$\nabla. \omega$$ and $$\nabla \times \omega$$ already defined (No. 54) are equivalent to $$\{ \nabla \omega \}_{\text{S}}$$ and $$\{ \nabla \omega \}_{\text{X}}.$$

160. An important case is that in which the vector operated on is of the form $$\nabla u.$$ We have then where  This dyadic, which is evidently self-conjugate, represents the six