Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/53

at the point considered above the average of its values at six points at the following vector distances: $ai,-ai,aj,-aj,ak,-ak.$ Since the directions of $i,j,$ and $k$ are immaterial (provided that they are at right angles to each other), the excess of the value of $u$ at the central point above its average value in a spherical surface of radius a constructed about that point as the center will be represented by the same expression, $-{\tfrac {1}{6}}a^{2}\nabla .\nabla u.$

Precisely the same is true of a vector function, if it is understood that the additions and subtractions implied in the terms average and excess are geometrical additions and subtractions.

Maxwell has called $-\nabla .\nabla u$ the concentration of $u,$ whether $u$ is scalar or vector. We may call $\nabla .\nabla u$ (or $\nabla .\nabla \omega$), which is proportioned to the excess of the average value of the function in an infinitesimal spherical surface above the value at the center, the dispersion of $u$ (or $\omega$).

Transformation of Definite Integrals.

73. From the equations of No. 65, with the principles of integration of Nos. 57, 59, and 60, we may deduce various transformations of definite integrals, which are entirely analogous to those known in the scalar calculus under the name of integration by parts. The following formulæ (like those of Nos. 67, 59, and 60) are written for the case of continuous values of the quantities (scalar and vector) to which the signs $\nabla ,\nabla .,$ and $\nabla \times$ are applied. It is left to the student to complete the formulæ for cases of discontinuity in these values. The manner in which this is to be done may in each case be inferred from the nature of the formula itself. The most important discontinuities of scalars are those which occur at surfaces: in the case of vectors discontinuities at surfaces, at lines, and at points, should be considered.

74. From equation (3) we obtain

$\int \nabla (tu).d\rho =t''u''-t'u'=\int u\nabla t.d\rho +\int t\nabla u.d\rho ,$

where the accents distinguish the quantities relating to the limits of the line-integrals. We are thus able to reduce a line-integral of the form $\int u\nabla t.d\rho$ to the form $-\int t\nabla u.d\rho$ with quantities free from the sign of integration.

76. From equation (5) we obtain

$\iint \nabla \times (u\omega ).d\sigma =\int u\omega .d\rho =\iint u\nabla \times \omega .d\sigma -\iint \omega \times \nabla u.d\sigma ,$

where, as elsewhere in these equations, the line-integral relates to the boundary of the surface-integral.

From this, by substitution of $\nabla t$ for $\omega ,$ we may derive as a particular case

$\iint \nabla u\times \nabla t.d\sigma =\int u\nabla t.d\rho =-\int t\nabla u.d\rho .$