Editing Cotangent space (section)

==The differential of a function==

Let <math>M</math> be a smooth manifold and let <math>f\in C^\infty(M)</math> be a [[smooth function]]. The differential of <math>f</math> at a point <math>x</math> is the map
:<math>\mathrm d f_x(X_x) = X_x(f)</math>
where <math>X_x</math> is a [[Differential geometry of curves|tangent vector]] at <math>x</math>, thought of as a derivation. That is <math>X(f)=\mathcal{L}_Xf</math> is the [[Lie derivative]] of <math>f</math> in the direction <math>X</math>, and one has <math>\mathrm df(X)=X(f)</math>.  Equivalently, we can think of tangent vectors as tangents to curves, and write
:<math>\mathrm d f_x(\gamma'(0))=(f\circ\gamma)'(0)</math>
In either case, <math>\mathrm df_x</math> is a linear map on <math>T_xM</math> and hence it is a tangent covector at <math>x</math>.

We can then define the differential map <math>\mathrm d:C^\infty(M)\to T_x^*(M)</math> at a point <math>x</math> as the map which sends <math>f</math> to <math>\mathrm df_x</math>. Properties of the differential map include:

# <math>\mathrm d</math> is a linear map: <math>\mathrm d(af+bg)=a\mathrm df + b\mathrm dg</math> for constants <math>a</math> and <math>b</math>,
# <math>\mathrm d(fg)_x=f(x)\mathrm dg_x+g(x)\mathrm df_x</math>

The differential map provides the link between the two alternate definitions of the cotangent space given above. Since for all <math> f \in I^2_x </math> there exist <math>g_i, h_i \in I_x</math> such that <math display="inline">f=\sum_i g_i h_i</math>, we have,
 
<math display>
\begin{array}{rcl}
\mathrm d f_x & = & \sum_i \mathrm d (g_i h_i)_x \\
 & = & \sum_i (g_i(x)\mathrm d(h_i)_x+\mathrm d(g_i)_x h_{i}(x)) \\
 & = & \sum_i (0\mathrm d(h_i)_x+\mathrm d(g_i)_x 0) \\
 & = & 0
\end{array} </math> 

So that all function in <math>I^2_x </math> have differential zero, it follows that for every two functions <math>f \in I^2_x</math>, <math>g \in I_x</math>, we have <math>\mathrm d (f+g)=\mathrm d (g)</math>. We can now construct an [[isomorphism]] between <math>T^*_x\!\mathcal M</math> and <math>I_x/I^2_x</math> by sending linear maps <math>\alpha</math> to the corresponding cosets <math>\alpha + I^2_x</math>. Since there is a unique linear map for a given kernel and slope, this is an isomorphism, establishing the equivalence of the two definitions.