Suppose, I have a multivariable function: $f(x, y) = 2x + 10y + 5$.
The gradient vector of this function would be:
$ \text{gradient of } f = \nabla f = \begin{bmatrix}\frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y}\end{bmatrix}$
What does a gradient vector represent if $f$ is considered as a surface in 3D?
As far as I understand:
A column vector of values:
1. instantaneous change of the surface represented by f in the x-direction,
2. instantaneous change of the surface represented by f in the y-direction
So, what is the intuition here if x represents, say, force, and y represents time?
Also, how does the intuition change if the number of variables increases?