What is meant by linearity in more than one dimension? I am slightly unclear on some terminology that I have come across in the following question:

Is $\partial _vf(0,0)$ linear in $v$?

My understanding of linearity for a single variable function is fine, although I'm a little unclear on what it means for a function that takes multiple variables? What is the generalisation here, and how should I interpret this word in these types of problems?
I would be grateful for any clarification here regarding the meaning of linearity in this context.

Hint: how does $\partial_{v+w} $ depend on $\partial_{v} + \partial_{w} $? The question makes sense since $v$ and $w$ are vectors you can add together.
The generalization from one variable to many is to think of the many variables collected into a single vector. So you are dealing with a function of one vector variable.
