Cotangent Space - The Differential of A Function

The Differential of A Function

Let M be a smooth manifold and let f ∈ C∞(M) be a smooth function. The differential of f at a point x is the map

df_x(X_x) = X_x(f)

where X_x is a tangent vector at x, thought of as a derivation. That is is the Lie derivative of f in the direction X, and one has df(X)=X(f). Equivalently, we can think of tangent vectors as tangents to curves, and write

df_x(γ′(0)) = (f o γ)′(0)

In either case, df_x is a linear map on T_xM and hence it is a tangent covector at x.

We can then define the differential map d : C∞(M) → T_x*M at a point x as the map which sends f to df_x. Properties of the differential map include:

1. d is a linear map: d(af + bg) = a df + b dg for constants a and b,
2. d(fg)_x = f(x)dg_x + g(x)df_x,

The differential map provides the link between the two alternate definitions of the cotangent space given above. Given a function f ∈ I_x (a smooth function vanishing at x) we can form the linear functional df_x as above. Since the map d restricts to 0 on I_x2 (the reader should verify this), d descends to a map from I_x / I_x2 to the dual of the tangent space, (T_xM)*. One can show that this map is an isomorphism, establishing the equivalence of the two definitions.