I am trying to visualize$\ z = f(x(t), y(t))$, and my model is that$\ x, y, z$ depend on$\ t$. The only way that I can visualize this is as a point moving in 3D space. However, wouldn't that be $\ \vec p = \vec f(t)$ ?

Is there any way to visualize the relationship between$\ z$ and$\ t$ as a surface that is changing in time, or is this wrong?

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    $\begingroup$ You're not missing something. Both of these representations can work. $\endgroup$ – David G. Stork Apr 1 at 17:36
  • $\begingroup$ Sorry David I realized that I had a more specific question instead of whether or not I was missing something. It's just that whenever I see f(x, y) I think surface, so I updated my question. $\endgroup$ – Benjamin Smus Apr 1 at 17:55

No, you should draw one surface $z=f(x,y)$ and think of parametrized curves on that surface. That is, given the parametrized curve $(x(t),y(t))$ in the $xy$-plane, you look up to the surface and watch the particle moving along the surface.

You could indeed end up in $4$-space by including the $t$-axis, i.e., plotting $(t,x(t),y(t))$ and then moving into one more dimension with the plot $(t,x(t),y(t),f(x(t),y(t)))$ — still a curve on the surface $z=f(x,y)$.

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