Advanced Calculus fi..

Chapter 3 Vector Differential Calculus 199In tensor analysis, one'allows changes of coordinates that are much more generalthan changes of scale. It is more convenient to number our coordinates as (x' , x2, x3)and simply to refer to the (x') coordinates. We can in fact reason generally aboutn-dimensional space and allow i to go from 1 to n. The use of superscripts ratherthan subscripts is required for a certain consistency of all the tensor notations. If weintroduce new coordinates (xi), then we have equationsrelating new and old coordinates. We consider these equations only in a neighborhoodD of a certain point and assume that they define a one-to-one mapping withinverseXIr. i' =fi(xl, ..., xn), i = 1, ..., n, (3.73)and that all functions in (3.72) and (3.73) are differentiable, with continuous secondpartial derivatives and nonzero JacobianIt is essential for the theory of tensors that we allow all such changes of coordinates.In the following discussion it will be convenient to denote by ((I, . . . , 6") ajixed Cartesian coordinate system in En, in terms of which distance and angle aremeasured as usual. We denote by (xl, . . . , xn) and (z', . . . , 3") two other, generallycurvilinear, coordiante systems introduced as before in the neighborhood D of apoint, and related by Eqs. (3.72) and (3.73) to each other and related in similarfashion to the (6'). We shall refer to the (6') as standard coordinates.Now let a vector field be given in the chosen neighborhood. Then the vectors allhave sets of components in each coordinate system. For a contravariant vectorjeld(or briefly, contravariant vector) the components will be denoted in the (xi) coordinatesby (u' , . . . , un) (upper indices) and in the (Z') coordinates by (kl, . . . , fin).Furthermore, for any two such coordinate systems the components are to be relatedby the ruleThis rule is chosen to fit the case when the vector in question is a velocity vector:For then-&x-d ~ i n azi dxj n ajriOf=--.- - C -uj.dt ,=I j=1 axjFor a covariant vector we denote the components in the two systems of coordinatesby (u . . . , u,) and (2 ,, . . . , fin), respectively, and require that