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I do not know how to automatically commute partial derivatives on a scalar and simplify it. For example, if I have: $$\partial_a \partial_b \partial_c f- \partial_c \partial_a \partial_b f $$ where $f$ is a scalar function, the ToCanonical and Simplification command do not simplify this. I also use CommuteCovdsonScalar command, but it doesn't work as well. Could anybody help me?

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The recommended way to do something like this is to use SortCovDs, which may introduce Christoffel terms if needed:

In[6]:= PD[-a]@PD[-b]@PD[-c]@F[] - PD[-c]@PD[-a]@PD[-b]@F[] // SortCovDs
Out[6]= 0

If you want to automate SortCovDs for a given derivative, say PD in this case, use:

SortCovDsStart[PD]

Then this will automatically give zero (again, for general covariant derivatives this would produce Christoffel terms):

In[9]:= PD[-a]@PD[-b]@PD[-c]@F[] - PD[-c]@PD[-a]@PD[-b]@F[]
Out[9]= 0

To stop the automatic commutation of derivatives use

SortCovDsStop[PD]
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