This work aims at introducing some energy operators linked to Teager-Kaiser energy operator (TKEO) (Kaiser, 1990), its associated higher order versions and expanding them to multi-dimensional signals. These operators are very useful for analysing oscillatory signals with time-varying amplitude and frequency (AMFM). We first propose a new mathematical expression of these operators using directional derivatives along any n-D vector and Kronecker powers (proposition 1, section 3). This mathematical formulation allows us to extend to n-D case some properties of the classical TKEO such as tracking of AM envelope and instantaneous frequency of a multi-dimensional AM-FM signal. In addition, we have introduced a new scalar function using the directional derivative along a vector to recover the "sign" of the frequency components . Applications of this model to a local n-D AM-FM signal and the related demodulation errors are presented. To show the effectiveness and the robustness of the new class of operators in term of envelope and frequency tracking, results obtained on synthetic and real data are compared to multi-dimensional energy separation algorithm (Maragos and Bovik, 1995) and to our previously developed method (Salzenstein and Boudraa, 2009). Finally, the performances of these methods are investigated in the presence of an additive noise.