Can you compute Laplacian, divergence, curl for a function?

In my physics class, we are currently studying gradient, Laplacian, divergence, and curl, and we have a problem that states to compute all four of these (I.e., (1) gradient, (2) Laplacian, (3) divergence, and (4) curl) “as appropriate” for the given expressions.

Now I noticed that some of the expressions are vectors and some are not. I’ve been reading about divergence and curl and know somewhat how they apply to vectors, vector fields. But do they apply to functions as well?

Likewise, how can you take the partial derivative of a vector?

Of note, I understand that the gradient and curl can be zero but here I am talking not about one of these operations being zero but rather about possibly not being able to do it at all.

For example, the first problem is computing “as appropriate” the gradient, Laplacian, divergence and curl of x^2+ysinz. Is it possible to provide a hint as to which of these would be possible to do and why/how?

Finally, I did originally ask this question in the Physics forum but was advised to ask here instead.

Thanks very much!