Tangent maps and orientation

Try

At p=(1,2,2) on the sphere of radius three, the vector v=(2,−1,0) is tangent because p·v=0. A different tangent vector can be realized by a different curve through p, but all such velocities lie in the same two-dimensional plane.

The tangent plane is a vector space attached to a point, not the affine plane drawn through that point in a picture. Translating that vector space by p produces the familiar affine tangent plane. Keeping those two meanings distinct prevents errors when taking differentials of maps.

Predict

Predict whether changing the surface parameters changes the tangent plane. The coordinate basis changes, but an invertible coordinate Jacobian cannot change its span.

Predict whether every regular surface has a continuous global normal. Each chart has one, but those local choices may not agree around a loop. Orientation is a global compatibility issue.

Experiment

Compare three descriptions at the sphere point.

Description	Tangency condition or basis
Level-set differential	dF_p(v)=2p·v=0
Normal vector	p/3
Curve velocity	Derivative of a sphere curve through p

Check v=(2,−1,0) and w=(2,0,−1); both are tangent and independent. Their span is the tangent space. An arbitrary vector can be decomposed into a tangent part and a component along p/3 using orthogonal projection.

Understand

For a chart X, every vector aX_u+bX_v is realized as the velocity of X(u₀+at,v₀+bt). Conversely, express any smooth surface curve through the chart inverse and apply the chain rule. Its velocity is a linear combination of those same coordinate derivatives. Thus the tangent space can be defined without choosing a preferred parametrization.

Under a regular coordinate change, the new derivatives are combinations of X_u,X_v with an invertible Jacobian. Their spans agree. A smooth map between surfaces has differential defined by differentiating images of curve velocities; the chart chain rule ensures this is a well-defined linear map.

The unit normal associated to an ordered chart is N=(X_u×X_v)/||X_u×X_v||. Under a coordinate change the cross product is multiplied by the Jacobian determinant. Normalization removes its absolute magnitude, leaving precisely its sign. Positive coordinate changes preserve orientation; negative ones reverse it.

Apply

A global orientation is a compatible continuous choice of unit normal for a surface in R³. On a Möbius strip, following a local normal once around the central loop returns its negative. That sign reversal obstructs a single-valued continuous choice. Local normal formulas remain valid, but an argument using one global N must check orientability.

Repair “The differential is the same matrix in every chart.” The linear map is intrinsic; its matrix changes with the source and target bases. Coordinate invariance concerns the map and resulting geometric quantities, not literal equality of all coefficient arrays.

For optional depth, the tangent space to a regular level set F=c is the kernel of dF. Curve differentiation gives inclusion, and both spaces have dimension two, yielding equality. This combines the level-set theorem with linear algebra rather than assuming every vector in the kernel is automatically a curve velocity.

Continue

State the point, tangent basis and orientation whenever a matrix represents a geometric derivative. Continue to the first fundamental form to measure lengths and angles within these tangent spaces.