The meaning of the sum of partial derivatives of a scalar function

Question

I have a real valued function $f(x_1,\dots,x_n)$ and it is important for me to add the coordinates of his gradient. In other words, if $u$ is the vector $(1, \dots, 1)$ I want to evaluate: $$\operatorname{div}(u \cdot f)$$ However strictly speaking this is not the divergence of $f$ since $f$ is not a vector field. Has anybody any idea if this corresponds to something specific in mathematics or physics? Can I, by abuse of language say that I have the "divergence" of a scalar function?

Answer 1

The sum of first-order partial derivatives has the meaning of directional derivative along the vector $u=(1,1,\dots,1)$. In physical terms, this is the rate of change of $f$ observed by someone moving with velocity $u$.

In some texts, the notion of directional derivatives is restricted to unit vectors; if so, then we are looking at $\sqrt{n}$ times the directional derivative along the unit vector $(1/\sqrt{n},\dots, 1/\sqrt{n})$.