Let \(u\) be a smooth real-valued function on a [smooth manifold](smooth manifold.md) \(M\). Then the \(k\)-th covariant derivative \(\nabla^ku\) is a \((0,k)\)-tensor field. In local coordinates we write \(u_{i_1\ldots i_k}\) as an abbreviation of \(u_{;i_1\ldots i_k}\).
Remarks
- For \(k=2\) and \(\nabla\) being the Levi-Civita connection the tensor \(\nabla^2 u\) is symmetric (see covariant Hessian ).
- For \(k>2\) the tensor \(\nabla^k u\) is not symmetric in all arguments in general (even if consider the Levi-Civita connection ). But still it is symmetric in the first two arguments, because of the previous remark.