Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/92

76 differential coefficients of the second order of $$u$$ with respect to $$x, y,$$ and $$z.$$

161. The operators $$\nabla \times$$ and $$\nabla .$$ may be applied to dyadics in a manner entirely analogous to their use with scalars. Thus we may define $$\nabla \times \Phi$$ and $$\nabla. \Phi$$ by the equations Then, if  Or, if  162. We may now regard $$\nabla. \nabla$$ in expressions like $$\nabla. \nabla \omega$$ as representing two successive operations, the result of which will be in accordance with the definition of No. 70. We may also write $$\nabla. \nabla \Phi$$ for although in this case we cannot regard $$\nabla. \nabla$$ as representing two successive operations until we have defined $$\nabla \Phi.$$ That $$\nabla. \nabla \Phi = \nabla \nabla. \Phi - \nabla \times \nabla \times \Phi$$ will be evident if we suppose $$\Phi$$ to be expressed in the form $$\alpha i + \beta j + \gamma k.$$ (See No. 71.)

163. We have already seen that where $$u'$$ and $$u''$$ denote the values of $$u$$ at the beginning and the end of the line to which the integral relates. The same relation will hold for a vector; i.e.,