Computation - can you compute the gradient, Laplacian, divergence and curl of any function?

Question

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.

Young Kindaichi · Answer

But do they apply to functions as well?

No!
Divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. The divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
The curl is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation.

how can you take the partial derivative of a vector?

I would rather say vector field (or space-dependent vector (maybe time too)). For example :
$$mathbf{F}=xhat i +yhat j+z hat k$$
$$frac{partial mathbf{F}}{partial x}=hat i$$
But there is no use in it.

about possibly not being able to do it at all.

If the vector field and its derivative are well defined at a point then you can compute a well-defined value divergence and curl at that point. But It's not always the case for example
$$mathbf{V}=frac{1}{r^2}hat{r}$$
has a singularity at origin thus you can not find the divergence just by differentiating.

Computation - can you compute the gradient, Laplacian, divergence and curl of any function?

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