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

We then introduce a function f(r,θ) and show its graph (r,θ,f(r,θ)) in the 3D graph window. The image of (r(t),θ(t)) under f produces a green curve (r(t),θ(t),z(t)) that runs along the surface of the graph. The curve (r(t),θ(t),z(t)) has its own velocity vector (r'(t),θ'(t),z'(t)) = (r'(t),θ'(t),f_r(r(t),θ(t))r'(t) + f_θ(r(t),θ(t))θ'(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 r- and θ-directions. Observe that these three vectors are essentially an orthogonal projection of the three vectors in the domain onto the tangent plane at (r(t₀),θ(t₀),f(r(t₀),θ(t₀)).

Clicking the checkbox will display tangent vectors calculated not using the chain rule to illustrate the importance of the chain rule.