Ivancevic_Applied-Diff-Geom

Technical Preliminaries: Tensors, Actions and Functors 71In case of a higher–dimensional manifold M, the situation is naturallymore complex, but the main structure of the Jacobi equation remains similar,D 2 ξ+ R(u, ξ, u) = 0,ds2 where D denotes the covariant derivative and R(u, ξ, u) is the curvaturetensor, a three–slot linear machine. In components defined in a local coordinatechart (x i ) on M, this equation readsD 2 ξ ids 2+ dx j dx lRi jklds ξk ds = 0,where Rjkl i are the components of the Riemann curvature tensor.2.1.4.2 Exterior Differential FormsRecall that exterior differential forms are a special kind of antisymmetricalcovariant tensors (see, e.g., [De Rham (1984); Flanders (1963)]). Suchtensor–fields arise in many applications in physics, engineering, and differentialgeometry. The reason for this is the fact that the classical vectoroperations of grad, div, and curl as well as the theorems of Green, Gauss,and Stokes can all be expressed concisely in terms of differential forms andthe main operator acting on them, the exterior derivative d. Differentialforms inherit all geometrical properties of the general tensor calculus andadd to it their own powerful geometrical, algebraic and topological machinery(see Figures 2.2 and 2.3). Differential p−forms formally occur asintegrands under ordinary integral signs in R 3 :• a line integral ∫ P dx + Q dy + R dz has as its integrand the one–formω = P dx + Q dy + R dz;• a surface integral ∫∫ A dydz + B dzdx + C dxdy has as its integrand thetwo–form α = A dydz + B dzdx + C dxdy;• a volume integral ∫∫∫ K dxdydz has as its integrand the three–formλ = K dxdydz.By means of an exterior derivative d, a derivation that transformsp−forms into (p + 1)−forms, these geometrical objects generalize ordinaryvector differential operators in R 3 :• a scalar function f = f(x) is a zero–form;