explanation of slope in a graph?

Question

I have been reading some text, and I have a problem to interpret the following figure:

$$ D(x)=[\delta s (y,x)/\delta y]y=x $$

It says that $D(x)$ is the slope against the $y$ direction, what does it mean exactly? I cannot see the interpretation.

Answer 1

Pick an $x$ value and slice the surface perpendicular to the $x$-axis at that $x$ value, generating a curve in the $y$-$z$ plane.

The partial derivative, ${\partial s\over \partial y}$, is the slope of that curve (in the single variable calculus sense of "slope of the tangent line").

You might find the explanation and pictures here helpful.