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.

Let $\mathrm D_{\boldsymbol u}=\boldsymbol u\cdot \nabla$ be a directional derivative. Let $f:\Bbb R^n\to\Bbb R$ be some differentiable function and choose some point $\boldsymbol x\in\Bbb R^n$. We now define a map $$\Phi:\Bbb R^n\to\Bbb R \\ \Phi:\boldsymbol u\mapsto (\mathrm D_{\boldsymbol u}f)(\boldsymbol x)$$ The directional derivative is linear in the sense that the map $\Phi$ is linear.
