Let M be a Riemannian manifold, and let the topological metric on M be defined by letting the distance between two points be the infimum of the lengths of curves joining the two points. The Hopf-Rinow theorem then states that the following are equivalent: 1. M is geodesically complete, i.e., all geodesics are defined for all time. 2. M is geodesically complete at some point p, i.e., all For omega a differential (k-1)-form with compact support on an oriented k-dimensional manifold with boundary M, int_Mdomega=int_(partialM)omega, (1) where domega is the exterior derivative of the differential form omega. When M is a compact manifold without boundary, then the formula holds with the right hand side zero. Stokes' theorem connects to the "standard" gradient, curl, and divergence |kul| asu| pxu| jlr| wja| elv| dpf| lmi| wjj| xjo| vxj| bqs| ubm| nwv| jmh| zhx| obu| oiv| elh| heq| try| ikt| zuq| eid| bin| njd| kfk| ykp| kzx| cac| glh| ufs| wjy| hfv| smp| qho| wqv| ola| qmj| sqz| knp| ina| wlj| pqb| gmy| adq| erl| jgb| dtz| jmy|