= Tangent space by point derivations
{title2=$T_xM=\operatorname{Der}_x(C^\infty_x(M),\mathbb R)$}
A tangent vector is a <linear map> on <germs> of smooth functions satisfying $D(ab)=a(x)D(b)+b(x)D(a)$. In a <manifold chart>, first-order expansion shows $Dh=\sum_iD(u^i)\partial_i h(x)$. Thus the coordinate <derivations> form a <basis>, and the <dimension> equals the manifold <dimension>. Composition of <germs> with a <smooth map> defines its <differential of a smooth map> without choosing charts.
Back to article page
