Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/69

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At Fig. 3 we have an illustration of curl combined with convergence.

A Treatise on Electricity and Magnetism Volume 1 Fig3.PNG
Fig. 3

Let us now consider the meaning of the equation

V \nabla \sigma = 0

This implies that \nabla \sigma is a scalar, or that the vector \sigma is the slope of some scalar function \Psi. These applications of the operator \nabla are due to Professor Tait[1]. A more complete development of the theory is given in his paper 'On Green's and other allied Theorems'[2] to which I refer the reader for the purely Quaternion investigation of the properties of the operator \nabla.

26.] One of the most remarkable properties of the operator \nabla is that when repeated it becomes

\nabla^2 = - ( \frac{d^2}{dx^2} +\frac{d^2}{dy^2} +\frac{d^2}{dz^2} )

an operator occurring in all parts of Physics, which we may refer to as Laplace's Operator.

This operator is itself essentially scalar. When it acts on a scalar function the result is scalar, when it acts on a vector function the result is a vector.

If, with any point P as centre, we draw a small sphere whose radius is r, then if q_0 is the value of q at the centre, and \bar{q} the mean value of q for all points within the sphere,


so that the value at the centre exceeds or falls short of the mean value according as \nabla^2 q is positive or negative.

I propose therefore to call \nabla^2q the concentration of q at the point P, because it indicates the excess of the value of q at that point over its mean value in the neighbourhood of the point.

If q is a scalar function, the method of finding its mean value is well known. If it is a vector function, we must find its mean value by the rules for integrating vector functions. The result of course is a vector.

  1. Proceedings R. S. Edin., 1862
  2. Trans. R. S. Edin., 1869-70.