Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/47

Or, if $\tau$ and $\omega$ are two single-valued continuous vector functions of any number of scalar or vector variables, and

	$d\tau =$	$d\omega ,$
then	$\tau =$	$\omega +\alpha ,$

where $\alpha$ is a vector constant.

When the above hypotheses are not satisfied in general, but will be satisfied if the variations of the independent variables are confined within certain limits, then the conclusions will hold within those limits, provided that we can pass by continuous variation of the independent variables from any values within the limits to any other values within them, without transgressing the limits.

49. So far, it will be observed, all operations have been entirely analogous to those of the ordinary calculus.

Functions of Position in Space.

60. Def.—If $u$ is any scalar function of position in space (ie., any scalar quantity having continuously varying values in space), $\nabla u$ is the vector function of position in space which has everywhere the direction of the most rapid increase of $u$, and a magnitude equal to the rate of that increase per unit of length. $\nabla u$ may be called the derivative of $u$, and $u$, the primitive of $\nabla u.$

We may also take any one of the Nos. 51, 52, 58 for the definition of $\nabla u.$

51. If $\rho$ is the vector defining the position of a point in space,

$du=\nabla u.d\rho .$

52.	$\nabla u=i{\frac {du}{dx}}+j{\frac {du}{dy}}+k{\frac {du}{dz}}\cdot$

53.	${\frac {du}{dx}}=i.\nabla u,$${\frac {du}{dy}}=j.\nabla u,$${\frac {du}{dz}}=k.\nabla u.$

54. Def.—If $\omega$ is a vector having continuously varying values in space,

	$\nabla .\,\omega =$	$i.{\frac {d\omega }{dx}}+j.{\frac {d\omega }{dy}}+k.{\frac {d\omega }{dz}},$	(1)
and	$\nabla \times \omega =$	$i\times {\frac {d\omega }{dx}}+j\times {\frac {d\omega }{dy}}+k\times {\frac {d\omega }{dz}}\cdot$	(2)

$\nabla .\,\omega$ is called the divergence of $\omega$ and $\nabla \times \omega$ its curl.

If we set

$\omega ={\text{X}}i+{\text{Y}}j+{\text{Z}}k,$

we obtain by substitution the equations

$\nabla .\,\omega ={\frac {d{\text{X}}}{dx}}+{\frac {d{\text{Y}}}{dy}}+{\frac {d{\text{Z}}}{dz}}$

and	$\nabla \times \omega =i\left({\frac {d{\text{Z}}}{dy}}-{\frac {d{\text{Y}}}{dz}}\right)+j\left({\frac {d{\text{X}}}{dz}}-{\frac {d{\text{Z}}}{dx}}\right)+k\left({\frac {d{\text{Y}}}{dx}}-{\frac {d{\text{X}}}{dy}}\right),$

which may also be regarded as defining $\nabla .\,\omega$ and $\nabla \times \omega .$