I’m looking for an equation that describes the components of the Laplacian of a general $(r,s)$ tensor over the real numbers. Basically all I know so far is that it should be a linear map

$$boldsymboltriangle:~^r_smathbb{R}to~^r_smathbb{R}$$

I.e, it preserves the order of the tensor. However, I can’t find any published formulae. Wolfram Mathworld writes the components of the tensor Laplacian using some very weird notation as

## $$left(boldsymboltriangle mathbf{T}right)^{i_1dots i_r}{}_{j_1dots j_s}=T^{i_1dots i_r}{}_{j_1dots j_s~;k}{}^{;k}$$

My question is simple: *what the heck is that?!* I know that the lowered semicolon index indicates a component of the covariant derivative. But what about the raised one? Is that the elusive contravariant derivative? Or is it instead a covariant derivative that has been "index raised" by contraction with the inverse metric? I’m afraid the short article on Mathworld is of little clarity and use (not to mention the readability is poor due to the low resolution of the typesetting). I would appreciate if someone could shed some light on this for me. Even only doing some special cases, i.e

- $(1,0)$ tensors (vectors)
- $(0,1)$ tensors (covectors)
- $(1,1)$ tensors (matrices)

Would be just fine as well. Thanks for the help.