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Usually, high order partial derivatives of a function $f:\mathbb{R}^n\to \mathbb{R}$ can be described by a multiindex $\mathbf{k}=(k_1,\dots, k_n)\in \mathbb{N}_0^n$, see for instance The quantity $k_i$ describes the number of partial derivatives in the $i$-th direction. The reason why this notation captures all cases of high order partial derivatives is that the order of differentiation is irrelevant, so that the number of derivatives in each coordinate completely describes a partial differential operator.

Suppose, we would like to describe high order partial covariant derivatives of a function $f:\mathbb{R}^n\to M$ for a Riemannian manifold $M$, for instance (n=2) $$\frac{D}{dx^1}\frac{D}{dx^2}\frac{d}{dx^1}f.$$

Since, in general, the order of differentiation is relevant, one cannot simply describe this operation by a multiindex $\mathbf{k}=(k_1,\dots, k_n)\in \mathbb{N}_0^n$.

For example, if we use the multiindex notation for the covariant derivative above, we would get the multiindex $(2,1)$, which would equally correspond to the operator $$\frac{D}{dx^2}\frac{D}{dx^1}\frac{d}{dx^1}f$$ which is different from the original covariant derivative operator.

My question: Does there exist a unified notation for high order partial covariant derivatives for functions $f:\mathbb{R}^d\to M$?

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What do you mean by "unified", unified with what? There's certainly notation for this -- just take the ordered analogue of multi-indices. So instead of using a free abelian monoid to index, you're using simply a free monoid. Is there anything more to your question? – Ryan Budney Mar 4 '13 at 9:08
No there is not anything more to the question. It just asks about the standard notation for partial covariant derivatives. – Philipp Mar 4 '13 at 9:19
I have never used multi-indices as unordered $n$-tuples. In most applications I am familiar with, i.e., tensor products of vector spaces and the related topic of higher covariant derivatives of tensors on a Riemannian manifold, the order matters. So for me a multi-index is always an ordered tuple. If you want some kind of (nonstandard) notation that distinguishes between the two, you might want to check Penrose's notation for tensors. – Deane Yang Mar 4 '13 at 14:15
Given an expression differentiated covariantly any number of times, the difference produced by an exchange of any pair of indices is an expression with fewer derivatives. In fact, this is how curvatures are usually introduced, as coefficients of those lower order terms. If you don't mind keeping derivatives of all orders in your expressions, rather than derivatives of a fixed order, you can symmetrize over all derivative indices (which can then be iterated over by multi-indices). Any differential expression can be written with such symmetrized derivatives. It is a way to parametrize jets. – Igor Khavkine Mar 4 '13 at 22:57

There is no standard notation. The problem is simply the noncommutativity, making the expressions unwieldy. It is rare in geometry to differentiate many times using a connection. Twice is usually enough.

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Occasionally three when you need the covariant derivative of curvature. – Deane Yang Mar 4 '13 at 22:30

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