Flows on a manifold

What you're seeing. Two views of the same one-parameter group of diffeomorphisms. On the left is the chart picture: the vector field v with components V^u = 0.85 + 0.15·v, V^v = 0.55·cos(0.7·u) on a flat plane — this is the (u, v)-coordinate representation. On the right the same data is lifted to the surface ℳ = { (u, v, z) : z = 0.5 cos 0.9u + 0.35 sin 0.6v } via Φ(u, v) = (u, v, h(u, v)). Field arrows on ℳ are pushforwards under dΦ; the orbit is the lifted integral curve; the gold tangent is v|_p = V^u∂_uΦ + V^v∂_vΦ.

Drag p on either panel: clicks on the manifold panel snap to the closest (u, v) via projection. The two views always show the same point.