Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/65

so that the ordinary conventions about the cyclic order of the symbols lead to a right-handed system of directions in space. Thus, if $x$ is drawn eastward and $y$ northward, $z$ must be drawn upward.

The areas of surfaces will be taken positive when the order of integration coincides with the cyclic order of the symbols. Thus, the area of a closed curve in the plane of $x,y$ may be written either

$\int x\ dy\qquad or\qquad -\int y\ dx$;

the order of integration being $x,y$ in the first expression, and $y,x$ in the second.

This relation between the two products $dxdy$ and $dydx$ may be compared with that between the products of two perpendicular vectors in the doctrine of Quaternions, the sign of which depends on the order of multiplication, and with the reversal of the sign of a determinant when the adjoining rows or columns are exchanged.

For similar reasons a volume-integral is to be taken positive when the order of integration is in the cyclic order of the variables $x,y,z,$ and negative when the cyclic order is reversed.

We now proceed to prove a theorem which is useful as establishing a connexion between the surface-integral taken over a finite surface and a line-integral taken round its boundary.

24.] THEOREM IV. A line-integral taken round a closed curve may be expressed in terms of a surface-integral taken over a surface bounded by the curve.

Let $X,Y,Z$ be the components of a vector quantity ${\mathfrak {A}}$ whose line-integral is to be taken round a closed curve s.

Let $S$ be any continuous finite surface bounded entirely by the closed curve $s$, and let $\xi ,\eta ,\zeta$ be the components of another vector quantity ${\mathfrak {B}}$, related to $X,Y,Z$ by the equations

$\xi ={\frac {dZ}{dy}}-{\frac {dY}{dz}}\qquad \eta ={\frac {dX}{dz}}-{\frac {dZ}{dx}}\qquad \zeta ={\frac {dY}{dx}}-{\frac {dX}{dy}}$

Then the surface-integral of ${\mathfrak {B}}$ taken over the surface $S$ is equal to the line-integral of ${\mathfrak {A}}$ taken round the curve $s$. It is manifest that $\xi ,\eta ,\zeta$ fulfil of themselves the solenoidal condition

${\frac {d\xi }{dx}}+{\frac {d\eta }{dy}}+{\frac {d\zeta }{dz}}=0$

Let $l,m,n$ be the direction-cosines of the normal to an element