differential_geometry.symbolic.compute_gradient#

differential_geometry.symbolic.compute_gradient(scalar_field: Basic, coordinate_axes: Sequence[Symbol], basis: Literal['unit', 'covariant', 'contravariant'] = 'covariant', inverse_metric: ImmutableDenseMatrix | MutableDenseMatrix | ImmutableDenseNDimArray | MutableDenseNDimArray | None = None) → MutableDenseNDimArray | ImmutableDenseNDimArray | ImmutableDenseMatrix | MutableDenseMatrix[source]#

Compute the symbolic gradient of a scalar field \(\phi\) in either covariant or contravariant basis.

Parameters:

  • scalar_field (Basic) – The scalar field \(\phi\) to differentiate. This should be any valid sympy expression dependent on the coordinate_axes and any other relevant symbols.

  • coordinate_axes (list of Symbol) – The coordinate axes (variables) with respect to which to compute the gradient. This should be the full list of the coordinate axes for the relevant coordinate system.

  • basis ('covariant' or 'contravariant', optional) –

    The basis in which to return the gradient. Defaults to ‘covariant’.

    Note

    if basis != 'covariant', the index must be raised and the inverse_metric will be used for contraction. If inverse_metric is not specified, an error results.

  • inverse_metric (MutableDenseMatrix or MutableDenseNDimArray) – Either a full inverse metric \(g^{\mu\nu}\) (shape (n, n)) or a 1D diagonal array of shape (n,) for orthogonal coordinates.

Returns:

The components of the gradient of \(\phi\), in the chosen basis.

Return type:

MutableDenseNDimArray

See also

compute_divergence, compute_laplacian

Examples

Compute the gradient of the scalar field \(\phi(r,\theta) = r^2 \sin(\theta)\).

>>> # Import the necessary functions.
>>> import sympy as sp
>>> from pymetric.differential_geometry.symbolic import compute_gradient
>>>
>>> # Create the symbols.
>>> r, theta, phi = sp.symbols('r theta phi')
>>> Phi = (r**2)*sp.sin(theta)
>>> inv_metric = sp.Matrix([[1,0,0],[0,r**2,0],[0,0,r**2*sp.sin(theta)]]).inv()
>>>
>>> # Compute the covariant gradient.
>>> compute_gradient(Phi, [r,theta, phi])
[2*r*sin(theta), r**2*cos(theta), 0]
>>>
>>> # Compute the contravariant gradient.
>>> compute_gradient(Phi, [r, theta, phi],basis='contravariant',inverse_metric=inv_metric)
[2*r*sin(theta), cos(theta), 0]
On this page

This Page