In single variable scalar function $\ f(x)\ $the sign of the derivative can tell you whether the function is increasing or decreasing at the point. I was trying to find an analogous concept in multi-variable scalar function $\varphi(\vec r)\ $since its output is a scalar quantity just like in the single variable function. Now in these functions we have the gradient vectors $\nabla\varphi(\vec a)\ $which gives us the direction of maximum rate of change of the scalar field at $\vec a $. But since it is a vector I assume its magnitude is always +ve and does not tell us anything about increase or decrease of the scalar quantity. Then there is the directional derivative $\varphi'(\vec a,\hat n) $ which is the dot product of the gradient vector $\nabla\varphi(\vec a)\ $and the the unit vector $\hat n $. In these dot products we get get scalar quantity with a sign. I used to believe that the sign indicated whether the function increased or decreased in that particular direction. But by definition of directional derivative $$\varphi'(\vec a,\hat n)=\nabla\varphi(\vec a)\cdot\hat n=|\nabla\varphi(\vec a)|\cos\theta $$ where $\theta$ is the angle betweenn $\hat n$ and $\nabla\varphi $. So as we can see the sign only arises from $\cos\theta $. So I assume the +ve sign indicates that the change along $\hat n $ is in the same direction as $\nabla\varphi $ i.e it is increasing if $\nabla\varphi $ is increasing or vice versa and the -ve sign indicates it is in the opposite direction of $\nabla \varphi $ i.e it increases when $\nabla\varphi\ $ decrease or vice versa.

So am I wrong or is there such an analogue ?