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

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

The scalar part is

and the vector part is

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

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

and

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 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  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 $$P$$.