Page:The Meaning of Relativity - Albert Einstein (1922).djvu/95

is Euclidean (more generally, if by a suitable choice of co-ordinates the $g_{\mu \nu }$ are constants) then the vector obtained at $G$ as a result of this displacement does not depend upon the choice of the curve joining $P$ and $G$. But otherwise, the result depends upon the path of the displacement. In this case, therefore, a vector suffers a change, $\Delta A^{\mu }$ (in its direction, not its magnitude), when it is carried from a point $P$ of a closed curve, along the

curve, and back to $P$. We shall now calculate this vector change:

$\Delta A^{\mu }=\int _{0}\delta A^{\mu }.$

As in Stokes' theorem for the line integral of a vector around a closed curve, this problem may be reduced to the integration around a closed curve with infinitely small linear dimensions; we shall limit ourselves to this case.