General Partial Derivatives in Higher-Dimensional Systems

The extension of vector-algebraic techniques to multicomponent systems of higher dimensionality (degrees of freedom / 2) can be carried out straightforwardly, even though one loses the convenience of mutually complementary pairs (X, X ) and orthogonal complementary conjugates (X, x ) that are a special feature of /= 2. In a space of / dimensions, a 

As before, we assume that the nonsingular metric matrix M of order/is known in terms of a chosen basis set / with conjugates / , such that 

We furthermore assume that the variables X, Y, Z, of the chosen V are known (i.e., given by expansions) in terms of the basis variables / or Rt for example, X is assumed to be given in either of the vector forms 

To sketch the general strategy for evaluating derivatives in multidimensional geometry, we first note that the derivative V in (12.41) would become rather simple if we had made a shrewd choice of basis variables. Specifically, if new basis variables 7 / (with conjugates R/ ) were chosen such that 

To achieve the desired simple form (12.47), it is only necessary to transform from the old conjugate basis Rt to the special conjugate basis R/ with the appropriate transformation matrix A that satisfies 

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GENERAL PARTIAL DERIVATIVES IN HIGHER-DIMENSIONAL SYSTEMS 405... 

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