I was reading a book (Nonlinear Elliptic Equations of the Second Order by Qin Han), and there was something which I didn't really understand. I will try to simply the setting for my question.
Let $a\in \mathbb R^n \setminus \{0\}.$
We consider the function $\ell(x)=M - a\cdot x$, where $M$ is a positive constant and $x\in \mathbb R^n$.
Then the graph of $\ell$ shall be a hyperplane (in $\mathbb R^{n+1}$) passing through the point $(0,M)$, where $0\in \mathbb R^n$.
The author wrote that, the graph of $\ell$ has the slope $a$, which I didn't understand.
I have two questions.
Q1: I looked up on MSE and someone said that the gradient is just the slope (of a plane), cf. https://math.stackexchange.com/a/712763/507382. Is this the definition of the slope of a plane? Why does it make sense geometrically?
Q2: Intuitively, $\nabla \ell= -a$, and the graph of $\ell$ shall have the slope $-a$, which is not $a$ as he said?
Thanks for help.