Tangent and cotangent spaces

If a smooth curve passes through p at time zero, differentiate a smooth scalar function along the curve. This gives a linear operation on functions, obeying the product rule with values evaluated at p. Curves with identical such directional derivatives define the same tangent vector. Equivalently, a tangent vector is a derivation at p on smooth function germs. In local coordinates every tangent vector has a unique expansion in coordinate derivative operators; the tangent space has the manifold's dimension.

The coordinate operators are basis vectors, not coordinate values. On a two-dimensional chart, the expression $v=2\partial_x-\partial_y$ means that v sends a function f to twice its x derivative minus its y derivative at the chosen point. No length or angle has been assigned yet. Tangent vectors based at different points belong to different vector spaces; identifying them by subtracting their component lists requires extra structure or a chosen coordinate convention.

The cotangent space is the dual vector space of linear maps from the tangent space to the real numbers. Coordinate differentials $dx^i$ form its dual basis, giving one on the matching coordinate derivative and zero on the others. For a smooth f, the differential df takes a vector v to v(f). This is coordinate-independent even though its component formula changes. A covector is not a tiny displacement; it is a rule for measuring one.

Under a smooth invertible coordinate change y=y(x), vector components transform by the Jacobian of the new coordinates with respect to the old. Covector components transform by the inverse Jacobian. Their contraction therefore stays fixed. The transition must have an invertible Jacobian on the chart overlap. A singular change such as y=x² at x=0 cannot serve as a smooth coordinate replacement there, even if its graph seems harmless on one side of the origin.

For a scalar field, df is defined using only the smooth structure. A gradient requires an inner product g on each tangent space: it is the vector satisfying $g(\operatorname{grad}f,v)=df(v)$ for every v. In Euclidean Cartesian coordinates the metric matrix is the identity, hiding the distinction because the two component lists look the same. In other coordinates, or with an anisotropic metric, converting the covector df into a gradient vector uses the inverse metric.

This distinction matters in optimization and field theory. A directional derivative is invariant under a mere relabeling of coordinates, while a naive update that treats partial derivatives as ordinary vector components may depend on that labeling. A metric provides a declared way to convert changes in a function into directions of steepest ascent. It does not follow from topology alone, and introducing it should be an explicit modeling decision rather than an accidental consequence of how a diagram was drawn.