The green curve (x(t),y(t),0) is shown in the horizontal plane (2D graph window). At the point (x(t₀),y(t₀)) on this curve, three vectors are drawn: the purple tangent vector (x'(t₀),y'(t₀),0), the red x-component (x'(t₀),0,0), and the blue y-component (0,y'(t₀),0).

We then introduce a function f(x,y) and show its graph (x,y,f(x,y)) in the 3D graph window. The image of (x(t),y(t)) under f produces a green curve (x(t),y(t),z(t)) that runs along the surface of the graph. The curve (x(t),y(t),z(t)) has its own velocity vector (x'(t),y'(t),z'(t)) = (x'(t),y'(t),f_x(x(t),y(t))x'(t) + f_y(x(t),y(t))y'(t)) which is given by the chain rule. This tangent vector is drawn at the point z(t₀) on the surface as well as its components in the x- and y-directions. Observe that these three vectors are essentially an orthogonal projection of the three vectors in the domain onto the tangent plane at (x(t₀),y(t₀),f(x(t₀),y(t₀)).