Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/51

64. If $u$ or $\omega$ is a function of several scalar or vector variables which are themselves functions of the position of a single point, the value of $\nabla u$ or $\nabla .\omega$ or $\nabla \times \omega$ will be equal to the sum of the values obtained by making successively all but each one of these variables constant.

65. By the use of this principle we easily derive the following identical equations:

$\nabla (t+u)=\nabla t+\nabla u.$

(1)

$\nabla .(\tau +\omega )=\nabla .\tau +\nabla .\omega .$⁠$\nabla \times [\tau +\omega ]=\nabla \times \tau +\nabla \times \omega .$

(2)

$\nabla (tu)=u\nabla t+t\nabla u.$

(3)

$\nabla .(u\omega )=\omega .\nabla u+u\nabla .\omega .$

(4)

$\nabla \times [u\omega ]=u\times \nabla \omega -\omega \nabla \times u.$

(5)

$\nabla .[\tau \times \omega ]=\omega .\nabla \times \tau -\tau .\nabla \times \omega .$

(6)

The student will observe an analogy between these equations and the formulæ of multiplication. (In the last four equations the analogy appears most distinctly when we regard all the factors but one as constant.) Some of the more curious features of this analogy are due to the fact that the $\nabla$ contains implicitly the vectors $i,j$ and $k,$ which are to be multiplied into the following quantities.

Combinations of the Operators $\nabla ,\nabla .,$ and $\nabla \times .$

66. If $u$ is any scalar function of position in space,

$\nabla \times \nabla u=0,$

as may be derived directly from the definitions of these operators.

67. Conversely, if $\omega$ is such a vector function of position in space that

$\nabla \times \omega =0,$

$\omega$ is the derivative of a scalar function of position in space. This will appear from the following considerations: The line-integral $\int \omega .d\rho$ will vanish for any closed line, since it may be expressed as the surface-integral of $\nabla \times \omega .$ (No. 60.) The line-integral taken from one given point $P'$ to another given point $P''$ is independent of the line between the points for which the integral is taken. (For, if two lines joining tiie same points gave different values, by reversing one we should obtain a closed line for which the integral would not vanish.) If we set $u$ equal to this line-integral, supposing $P''$ to be variable and $P'$ to be constant in position, $u$ will be a scalar function of the position of the point $P'',$ satisfying the condition $du=\omega .d\rho ,$ or, by No. 51, $\nabla u=\omega .$ There will evidentily be an infinite number of functions satisfying this condition, which will differ from one another by constant quantities.