Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/68

multiplication of $i,j,k,$ we find that $\nabla \sigma$ consists of two parts, one scalar and the other vector.

The scalar part is

$S\nabla \sigma =-({\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}})$, see Theorem III,

and the vector part is

$V\nabla \sigma =i({\frac {dZ}{dy}}-{\frac {dY}{dz}})+j({\frac {dx}{dz}}-{\frac {dX}{dx}})+k({\frac {dY}{dx}}-{\frac {dX}{dy}})$.

If the relation between $X,Y,Z$ and $\xi ,\eta ,\zeta$ is that given by equation (1) of the last theorem, we may write

$V\nabla \sigma =i\xi +j\eta +k\zeta$. See Theorem IV.

It appears therefore that the functions of $X,Y,Z$ which occur in the two theorems are both obtained by the operation $\nabla$ on the vector whose components are $X,Y,Z$. The theorems themselves may be written

$\iiint S\nabla \sigma d\varsigma \ =\iint S.\sigma U\nu ds$, (III)

and

$\int S\sigma d\rho =\iint S.\nabla \sigma U\nu ds$; (IV)

where $d\varsigma$ is an element of a volume, $ds$ of a surface, $d\rho$ of a curve, and $U\nu$ a unit-vector in the direction of the normal. To understand the meaning of these functions of a vector, let us suppose that $\sigma _{0}$ is the value of $\sigma$ at a point $P$, and let us examine

Fig. 1

the value of $\sigma -\sigma _{0}$ in the neighbourhood of $P$.

If we draw a closed surface round $P$ then, if the surface-integral of $\sigma$ over this surface is directed inwards, $S\nabla \sigma$ will be positive, and the vector $\sigma -\sigma _{0}$ near the point $P$ will be on the whole directed towards $P$, as in the figure (1).

I propose therefore to call the scalar part of $\nabla \sigma$ the convergence of $\sigma$ at the point $P$.

To interpret the vector part of $\nabla \sigma$, let us suppose ourselves to be looking in the direction of the vector whose

Fig. 2

components are $\xi ,\eta ,\zeta ,$ and let us examine the vector $\sigma -\sigma _{0}$ near the point $P$. It will appear as in the figure (2), this vector being arranged on the whole tangentially in the direction opposite to the hands of a watch.

I propose (with great diffidence) to call the vector part of $\nabla \sigma$ the curl, or the version of $\sigma$ at the point $P$.