Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/59

 21.] If two curves are such that one of them may be transformed into the other by continuous motion without at any time passing through any part of space for which the condition of having a potential is not fulfilled, these two curves are called Reconcileable curves. Curves for which this transformation cannot be effected are called Irreconcileable curves

The condition that $$Xdx+Ydy+Zdz$$ is a complete differential of some function $$\Psi$$ for all points within a certain region, occurs in several physical investigations in which the directed quantity and the potential have different physical interpretations.

In pure kinematics we may suppose X, Y, Z to be the components of the displacement of a point of a continuous body whose original coordinates are x, y, z, then the condition expresses that these displacements constitute a non-rotational strain.

If X, Y, Z represent the components of the velocity of a fluid at the point x, y, z, then the condition expresses that the motion of the fluid is irrotational.

If X, Y, Z represent the components of the force at the point x, y, z, then the condition expresses that the work done on a particle passing from one point to another is the difference of the potentials at these points, and the value of this difference is the same for all reconcileable paths between the two points.

On Surface-Integrals.

21.] Let d,S be the element of a surface, and $$\epsilon$$ the angle which a normal to the surface drawn towards the positive side of the surface makes with the direction of the vector quantity R, then $$\iint R\ cos\ \epsilon\ dS$$ is called the surface-integral of R over the surface S.

THEOREM III. The surface-integral of the flux through a closed surface may be expressed as the volume-integral of its convergence taken within the surface. (See Art. 25.)

Let X, Y, Z' be the components of R, and let l, m, n, be the direction-cosines of the normal to S measured outwards. Then the surface-integral or R over S is

$$\iint R\ cos\ \epsilon\ dS=\iint XldS+\iint YmdS+\iint ZndS$$