tv_entries_derivatives
======================

.. py:function:: t3toolbox.backend.sampling_derivatives.tv_entries_derivatives(index, pp, variation, frame, order)

   .. code-block:: python

      def tv_entries_derivatives(
              index:      NDArray,                # int, shape=(d,)+W -- the grid points
              pp:         typ.Sequence[NDArray],  # perturbation vectors P, len=d, elm_shape=W+(Ni,)
              variation:  typ.Tuple[
                  typ.Sequence[NDArray],          # var_tucker_cores dU. len=d, elm_shape=K+C+(nOi,Ni)
                  typ.Sequence[NDArray],          # var_tt_cores     dG. len=d, elm_shape=K+C+(rLi,nUi,rRi)
              ],                                  # = T3Variations.data
              frame:       typ.Tuple[
                  typ.Sequence[NDArray],          # up_tucker_cores  U. len=d
                  typ.Sequence[NDArray],          # down_tt_cores    O. len=d
                  typ.Sequence[NDArray],          # left_tt_cores    P. len=d
                  typ.Sequence[NDArray],          # right_tt_cores   Q. len=d
              ],                                  # = T3Frame.data = (up, down, left, right) = (U, O, P, Q)
              order:      int,                    # highest derivative order
      ) -> NDArray:                               # entries-derivative jets, shape=(order+1,)+W+K+C


   Symmetric derivatives of an entry of a tangent vector, in one repeated direction.

   The ``entries`` analog of :py:func:`tv_apply_derivatives` -- identical but with the frame/var
   up-index jets from slicing Tucker-core fibers at ``index`` (order 0) + contracting ``P`` (order 1).
   Stacks ``order + W + K + C``. Verified vs :py:func:`dense_entries_derivatives` on the densified tangent.

   .. seealso:: :py:obj:`t3_entries_derivatives`, :py:obj:`tv_apply_derivatives`, :py:obj:`entries.tv_entries`