I'm having a hard time understanding the geometrical meaning of the gradient. I understand how to calculate it on theory, given a function. But what if we don't have the function, but just a 3d graphic with points like this one:
I assume if I am able to find the gradients in every corner point of every "rectangle", Ill easily get the whole gradient, but how can I do that? What exactly does the gradient in this case? Should I compare the areas of all neighbour rectangles to see where there's a slope? Should I somehow check in which direction the area changes?
Anything easily explained will be of great help to my understanding, thank you.
For a function $z=f(x,y)$ the gradient vector in $(x_0,y_0)$ is
$$\nabla f(x_0,y_0)=f_x(x_0,y_0)\,\vec i+f_y(x_0,y_0)\,\vec j$$
Note that it is defined in the plane $(x,y)$ and represent geometrically the direction with maximum slope of the graph $z=f(x,y)$ and the maximum slope is given by the norm i.e."lenght of the gradient" vector.
Note also that by the gradient we can calculate the directional derivative for every direction defined by a vector $\vec v$ by the doct product with the gradient, that is:
$$f_{\vec v}=\frac {\partial f}{\partial \vec v}=\nabla \cdot \vec v$$
Note that in your graph near the corner point gradient vector tends to vanish since the slope is almost zero.