Surface charts and regular values

Try

The graph X(u,v)=(u,v,u²+v²) is a smooth surface. Its coordinate tangent vectors are (1,0,2u) and (0,1,2v), which are independent everywhere. Projection onto the first two coordinates supplies a smooth inverse onto the parameter domain.

A surface chart requires more than a formula with three outputs. It should be a homeomorphism from an open subset of R² onto a surface neighborhood and have derivative rank two. A map with rank two is an immersion locally, but can wrap around or self-intersect globally.

Predict

Predict what the usual spherical coordinates do at a pole. The angular derivative vanishes there, so that particular formula loses rank. The sphere itself is still smooth, because another chart works.

Predict whether a vanishing gradient in one level-set equation proves the level set singular. It does not: the same smooth plane can be described by z³=0, whose gradient vanishes on it. A failed sufficient test is not a proof of failure.

Experiment

Audit three examples.

Description	Rank or gradient check	Conclusion
Graph z=u²+v²	Cross product (−2u,−2v,1)	Regular everywhere
Sphere x²+y²+z²=R², R>0	Gradient never zero on level	Regular surface
Double cone z²=x²+y² at zero	Gradient zero	Need a separate obstruction

For the graph, compute the squared cross-product magnitude 1+4u²+4v². For the cone, explain why the gradient test is inconclusive by itself before considering the topology of the vertex.

Understand

The implicit function theorem proves the regular-level-set criterion. If F:R³→R is smooth and ∇F(p)≠0 at a level point, at least one partial derivative is nonzero. Solve locally for that coordinate as a smooth function of the other two. The level set is therefore locally a graph with an explicit regular chart.

For a sphere of radius R>0, its gradient 2p is nonzero at every level point. Near the north pole, the graph z=√(R²−x²−y²) supplies the chart missing from the angular parametrization.

The double cone at zero has a different problem. Any sufficiently small neighborhood in the cone contains points on both the upper and lower nappes. Removing the vertex separates those parts because the sign of z cannot change along the cone without passing through zero. A punctured disk chart is connected. If a surface chart existed at the vertex, a sufficiently small disk in that chart would give a connected punctured neighborhood meeting both nappes, contradiction. This is a genuine obstruction rather than a failed derivative test.

Apply

Repair “The parametrization is smooth, so its image is a regular embedded surface.” Smoothness alone does not ensure rank, injectivity or the correct subspace topology. Check those conditions separately. A cylinder wrapping map is regular but not globally injective; a self-intersecting immersion can have multiple sheets at one image point.

For optional depth, abstract manifolds allow charts without placing the surface inside R³. The later intrinsic metric lessons naturally extend to that setting. The present surface calculations begin with embedded or locally immersed patches and say when a global normal or embedding is required.

Continue

For each surface example, provide either a valid chart with inverse or a regular-value argument, and distinguish a chart defect from a geometric singularity. Continue to tangent maps and orientation to show which derivative information is independent of coordinates.