differential_geometry.dense_ops.dense_vector_divergence_covariant_full#

differential_geometry.dense_ops.dense_vector_divergence_covariant_full(vector_field: ndarray, Dterm_field: ndarray, inverse_metric_field: ndarray, *varargs, field_axes: Sequence[int] | None = None, derivative_axes: Sequence[int] | None = None, edge_order: Literal[1, 2] = 2, out: ndarray | None = None, **kwargs) → ndarray[source]#

Compute the divergence of a covariant vector field in a general coordinate system using a full inverse metric tensor.

This function converts the covariant vector field to its contravariant form by contracting with the inverse metric, and then computes the divergence using:

\[\nabla_i V^i = D_i V^i + \partial_i V^i\]

where:

  • \(V^i = g^{ij} V_j\) is the contravariant form of the input covariant field.

  • \(D_i = (\partial_i \rho) / \rho\) is the logarithmic derivative of the volume density.

  • \(\nabla_i V^i\) is the full covariant divergence in curved space.

Parameters:

  • vector_field (numpy.ndarray) – Contravariant vector field with shape (F_1, ..., F_M, ndim), where the final axis corresponds to coordinate directions. The first m axes are spatial grid axes. Must be broadcast-compatible with Dterm_field.

  • Dterm_field (numpy.ndarray) – D-term array of shape (..., ndim), where the last axis matches the number of coordinate directions. Must broadcast with the spatial axes of vector_field.

  • inverse_metric_field (numpy.ndarray) – Inverse metric tensor with shape (…, N, N). This is used to raise the index of the covariant field.

  • *varargs –

    Grid spacing for each axis. Follows the same format as numpy.gradient(), and can be:

    • A single scalar (applied to all spatial axes),

    • A list of scalars (one per axis),

    • A list of coordinate arrays (one per axis),

    • A mix of scalars and arrays (broadcast-compatible).

    If derivative_axes is provided, then varargs must match its shape. Otherwise, there must be M elements in varargs.

  • field_axes (list of int, optional) – Maps each grid axis (0 to m-1) to a corresponding component index in the vector field. Defaults to identity mapping [0, 1, ..., m-1].

  • derivative_axes (list of int, optional) – Grid axes along which to compute partial derivatives. If not specified, all spatial axes listed in field_axes are used.

  • edge_order ({1, 2}, default 2) – Accuracy order of finite differences used in derivative computation.

  • out (numpy.ndarray, optional) –

    Optional output buffer into which the result is placed. Specifying out can help to conserve memory.

    out should be specified so that it is the broadcast shape of vector_field and Dterm_field excluding the final (component) dimension of each. Thus, if vector_field is (A,B,1,3) and Dterm_field is (1,B,C,3), out should be (A,B,C).

  • **kwargs – Additional keyword arguments passed to the metric contraction routine.

Returns:

Divergence of the covariant vector field, with shape vector_field.shape[:-1].

Return type:

ndarray

See also

dense_vector_divergence_contravariant

Performs divergence on contravariant fields.

dense_contract_with_metric

Raises the index of the vector field via contraction.