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I want to calculate the divergence of a rank-2 tensor field $$\nabla \cdot T$$ defined on the surface of a sphere in spherical coordinates. As an example, let the field be given as follows :

import numpy as np
np.random.seed(42)

LAT = 100
LONG = 100

tensor_field = np.array([np.random.rand(LAT,LONG) for i in range(9)])

here, each one of the 9 entries corresponds to one component in spherical coordinates of the rank-2 tensor.

If it were a rank-1 tensor field (a vector field), I could brute force my way by manually converting each component to Cartesian basis, and calculate the divergence as done in this answer.

Is there a simpler way for higher rank tensor fields?

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  • $\begingroup$ Welcome to Scicomp! Have you checked: en.wikipedia.org/wiki/… ? $\endgroup$
    – MPIchael
    Commented Jan 22 at 9:50
  • $\begingroup$ Have you tried to write down the contraction of the covariant derivative with the tensor in index notation? $\endgroup$
    – Bort
    Commented Jan 24 at 17:13

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