Page:The Meaning of Relativity - Albert Einstein (1922).djvu/89

Rh That there are, nevertheless, in the general case, invariant differential operations for tensors, is recognized most satisfactorily in the following way, introduced by Levi-Civita and Weyl. Let $$A^\mu$$ be a contra-variant vector whose components are given with respect to the co-ordinate system of the $$x_\nu$$. Let $$P_1$$ and $$P_2$$ be two infinitesimally near points of the continuum. For the infinitesimal region surrounding the point $$P_1$$, there is, according to our way of considering the matter, a co-ordinate system of the $$X_\nu$$ (with imaginary $$X_\nu$$-co-ordinates) for which the continuum is Euclidean. Let $$A_{(1)}^\mu$$ be the co-ordinates of the vector at the point $$P_1$$. Imagine a vector drawn at the point $$P_2$$, using the local system of the $$X_\nu$$, with the same co-ordinates (parallel vector through $$P_2$$, then this parallel vector is uniquely determined by the vector at $$P_1$$ and the displacement. We designate this operation, whose uniqueness will appear in the sequel, the parallel displacement of the vector $$A^\mu$$ from $$P_1$$ to the infinitesimally near point $$P_2$$  If we form the vector difference of the vector $$(A^\mu)$$ at the point $$P_2$$ and the vector obtained by parallel displacement from $$P_1$$ to $$P_2$$, we get a vector which may be regarded as the differential of the vector $$(A^\mu)$$ for the given displacement $$(dx_\nu)$$.

This vector displacement can naturally also be considered with respect to the co-ordinate system of the $$x_\nu$$. If $$A^\nu$$ are the co-ordinates of the vector at $$P_1$$, $$A^\nu + \delta A^\nu$$ the co-ordinates of the vector displaced to $$P_2$$ along the interval $$(dx_\nu)$$, then the $$\delta A^\nu$$ do not vanish in this case. We know of these quantities, which do not have a vector