Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/46

46. Differential coefficient with respect to a scalar.—The quotient obtained by dividing the differential of a vector due to the variation of any scalar of which it is a function by the differential of that scalar is called the differential coefficient of the vector with respect to the scalar, and is indicated in the same manner as the differential coefficients of ordinary analysis.

If we suppose the quantities occurring in the six equations of the last section to be functions of a scalar $t$, we may substitute ${\frac {d}{dt}}$ for $d$ in those equations since this is only to divide all terms by the scalar $dt$.

47. Successive differentiations.—The differential coefficient of a vector with respect to a scalar is of course a finite vector, of which we may take the differential, or the differential coefficient with respect to the same or any other scalar. We thus obtain differential coefficients of the higher orders, which are indicated as in the scalar calculus.

A few examples will serve for illustration.

If $\rho$ is the vector drawn from a fixed origin to a moving point at any time $t,{\frac {d\rho }{dt}}$ will be the vector representing the velocity of the point, and ${\frac {d^{2}\rho }{dt^{2}}}$ the vector representing its acceleration.

If $\rho$ is the vector drawn from a fixed origin to any point on a curve, and $s$ the distance of that point measured on the curve from any fixed point, ${\frac {d\rho }{ds}}$ is a unit vector, tangent to the curve and having the direction in which $s$ increases; ${\frac {d^{2}\rho }{ds^{2}}}$ is a vector directed from a point on the curve to the center of curvature, and equal to the curvature; ${\frac {d\rho }{ds}}\times {\frac {d^{2}\rho }{ds^{2}}}$ is the normal to the osculating plane, directed to the side on which the curve appears described counter-clockwise about the center of curvature, and equal to the curvature. The tortuosity (or rate of rotation of the osculating plane, considered as positive when the rotation appears counter-clockwise as seen from the direction in which $s$ increases) is represented by

${\frac {{\frac {d\rho }{ds}}\cdot {\frac {d^{2}\rho }{ds^{2}}}\times {\frac {d^{3}\rho }{ds^{3}}}}{{\frac {d^{2}\rho }{ds^{2}}}\cdot {\frac {d^{2}\rho }{ds^{2}}}}}\cdot$

48. Integration of an equation between differentials.—If $t$ and $u$ are two single-valued continuous scalar functions of any number of scalar or vector variables, and

	$dt=$	$du,$
then⁠	$t=$	$u+a,$

where $a$ is a scalar constant.