Drag the gold point and watch the coordinate-basis vectors re-orient themselves as the base point moves.
What you are looking at. The surface ℳ is the graph of
h(x,y) = 0.5 cos(0.9 x) + 0.35 sin(0.6 y) — a gentle two-bump landscape.
The chart ψ : O → U ⊂ ℝ² is just vertical projection (x, y, z) ↦ (x, y).
In the chart panel on the right, the standard basis vectors e1 = (1,0) and
e2 = (0,1) at ψ(p) are always the same — they don't depend on p.
The interesting bit. Their pre-images under ψ−1 —
the coordinate-basis tangent vectors ∂1|p and
∂2|p on the surface (left panel) — are
not the same at every point. They push forward to
Φ∗eμ = (δμ1, δμ2, ∂h/∂xμ),
so the vertical component picks up the local slope of the surface. As you drag p
across peaks, valleys, and saddles, each arrow tilts up or down accordingly.
Why this matters. The coordinate-basis vectors span the tangent space Tpℳ at each point, but the identification is tied to the chart — a different chart gives a different basis, related to this one by the chain-rule Jacobian. This is the origin of the (contravariant) vector transformation law.