Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/94

transformation of the expression which would be allowable if the $\nabla$ were a vector (viz., by changes in the order of the factors, in the signs of multiplication, in the parentheses written or implied, etc.), by which changes the $\nabla$ is brought into connection with one particular factor, the expression thus transformed will represent the part of the value of the original expression which results from the variation of that factor.

167. From the relations indicated in the last four paragraphs, may be obtained directly a great number of transformations of definite integrals similar to those given in Nos. 74–77, and corresponding to those known in the scalar calculus by the name of integration by parts.

168. The student will now find no difficulty in generalizing the integrations of differential equations given in Nos. 78–89 by applying to vectors those which relate to scalars, and to dyadics those which relate to vectors.

169. The propositions in No. 90 relating to minimum values of the volume-integral $\iiint u\omega .\omega \,dv$ may be generalized by substituting $\omega .\Phi .\omega$ for $u\omega .\omega ,\Phi$ being a given dyadic function of position in space.

170. The theory of the integrals which have been called potentials, Newtonians, etc (see Nos. 91–102) may be extended to cases in which the operand is a vector instead of a scalar or a dyadic instead of a vector. So far as the demonstrations are concerned, the case of a vector may be reduced to that of a scalar by considering separately its three components, and the case of a dyadic may be reduced to that of a vector, by supposing the dyadic expressed in the form $\phi i+\chi j+\omega k$ and considering each of these terms separately.

CHAPTER V.

concerning transcendental functions of dyadics.

171. Def.—The exponential function, the sine and the cosine of a dyadic may be defined by infinite series, exactly as the corresponding functions in scalar analysis, viz.,

${\begin{aligned}e^{\Phi }&={\text{I}}+\Phi +{\tfrac {1}{2}}\Phi ^{2}+{\tfrac {1}{2.3}}\Phi ^{3}+{\text{etc.,}}\\\sin \Phi &=\Phi -{\tfrac {1}{2.3}}\Phi ^{3}+{\tfrac {1}{2.3.4.5}}\Phi ^{5}-{\text{etc.,}}\\\cos \Phi &={\text{I}}-{\tfrac {1}{2}}\Phi ^{2}+{\tfrac {1}{2.3.4}}\Phi ^{4}-{\text{etc.}}\end{aligned}}$

These series are always convergent. For every value of $\Phi$ there is one and only one value of each of these functions. The exponential function may also be defined as the limit of the expression

$\left({\text{I}}-{\frac {\Phi }{\text{N}}}\right)^{\text{N}},$