# difference between $f(x,y)$ and $f(x(t),y(t))$?

I somehow grasp that the one graph ($$f(x,y)$$) is a surface and the other ($$f(x(t),y(t))$$) is a curve but not the manner in which they are plotted. As far as I understand, for the surface one I do calculate $$z = f(x,y)$$ and obtain a hight for every point $$(x,y)$$ that forms my surface. However, isn't the curvy one actually build the same way and thus should show up as a surface?

You didn't talked about the domain of the functions. Lets say, for the sake of simplicity, that $$f:[a,b]\times [c,d] \to \mathbb R$$. Then the graph of $$f(x,y)$$ is the set $$\{(x,y,f(x,y)) \in \mathbb R^3\mid x\in [a,b], y \in [c,d]\}$$. If the function is continuous and positive, its graph would be a surface above the rectangle $$[a,b]\times [c,d]$$
If you choose functions $$x:[0,1] \to [a,b]$$, $$y:[0,1] \to [c,d]$$ and again $$x$$ and $$y$$ are continuous functions, then $$(x(t), y(t))$$ with $$t \in [0,1]$$ describes a curve $$\Gamma$$ inside the rectangle $$[a,b]\times [c,d]$$. So the graph of $$f(x(t),y(t))$$ is the set $$\{(x(t),y(t),f(x(t),y(t))) \in \mathbb R^3\mid t \in [0,1]\}$$, those points are on the surface, but they are only the ones that are above $$\Gamma$$.
In the image, the rectangle (the domain of $$f$$) is yellow, $$\Gamma$$ is black, the graph of $$f(x,y)$$ is green, and the graph of $$f(x(t),y(t))$$ is red.
The two expressions mean the same thing. The latter just specifies what the variables $$x,y$$ depend on. In this case time. You can think of as $$x$$ and $$y$$ vary in time, they draw out the surface $$f(x,y)=f(x(t),y(t))$$ as $$t$$ varies. I hope this helps with your understanding :).