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

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24.]
LINE-INTEGRAL AND SURFACE-INTEGRAL.
25

so that the ordinary conventions about the cyclic order of the symbols lead to a right-handed system of directions in space. Thus, if is drawn eastward and northward, 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 may be written either

;

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

This relation between the two products and 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 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 be the components of a vector quantity whose line-integral is to be taken round a closed curve s.

Let be any continuous finite surface bounded entirely by the closed curve , and let be the components of another vector quantity , related to by the equations

Then the surface-integral of taken over the surface is equal to the line-integral of taken round the curve . It is manifest that fulfil of themselves the solenoidal condition

Let be the direction-cosines of the normal to an element