differential_geometry.symbolic#
Symbolic manipulations for differential geometry operations.
This module provides symbolic tools for computing geometric quantities such as gradients, divergences, Laplacians, and metric-related operations in general curvilinear coordinates using SymPy.
These operations are essential in symbolic tensor calculus, particularly in the context of differential geometry, general relativity, and coordinate transformations in physics and applied mathematics.
These functions are integrated intoPyMetric for advanced handling of geometry dependencies, such as computing specialized D- and L-terms.
Functions
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Adjust the variance signature of a symbolic tensor by raising or lowering indices as needed. |
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Compute the D-term components for a particular coordinate system from the metric density function. |
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Compute the L-term components for a general or orthogonal coordinate system from the metric density \(\rho\) and the inverse metric \(g^{\mu\nu}\). |
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Compute the divergence \(\nabla \cdot {\bf F}\) of a vector field symbolically. |
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Compute the symbolic gradient of a scalar field \(\phi\) in either covariant or contravariant basis. |
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Compute the Laplacian \(\nabla^2 \phi\) of a scalar field in general or orthogonal curvilinear coordinates. |
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Compute the metric density function \(\sqrt{\det(g)}\). |
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Compute the gradient of a symbolic tensor field with arbitrary rank. |
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Compute the Laplacian of each component of a symbolic tensor field. |
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Compute the inverse of the metric \(g_{\mu \nu}\). |
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Lower a single index of a tensor using the provided metric. |
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Raise a single index of a tensor using the provided inverse metric. |