Derivation at a point
= Derivation at a point
{title2=$v:C_p^\infty(M)\to\mathbb R$}
At $p$ on a <smooth manifold>, a derivation at that point is a real-linear map $v:C_p^\infty(M)\to\mathbb R$ on <germs> of smooth functions with $v(fh)=f(p)v(h)+h(p)v(f)$. It defines a <tangent vector>. The local factorization $f-f(p)=\sum_i(x^i-x^i(p))f_i$ gives $v(f)=\sum_i v(x^i)\partial_i f(p)$, so the coordinate derivations form a basis of the <tangent space>.