Differential (mathematics)
Differentials as germs of functions

This approach works on any differentiable manifold. If

then f is equivalent to g at p, denoted f ∼ p g {\displaystyle f\sim _{p}g}, if and only ifthere is an open W ⊆ U ∩ V {\displaystyle W\subseteq U\cap V} containing p such that f (x) = g (x) {\displaystyle f(x)=g(x)} for every x in W.The germ of f at p, denoted p {\displaystyle _{p}}, is the set of all real continuous functions equivalent to f at p; if f is smooth at p then p {\displaystyle _{p}} is a smooth germ.If

then

This shows that the germs at p form an algebra.

Define I p {\displaystyle {\mathcal {I}}_{p}} to be the set of all smooth germs vanishing at p and I p 2 {\displaystyle {\mathcal {I}}_{p}^{2}} to be the product of ideals I p I p {\displaystyle {\mathcal {I}}_{p}{\mathcal {I}}_{p}}. Then a differential at p (cotangent vector at p) is an element of I p / I p 2 {\displaystyle {\mathcal {I}}_{p}/{\mathcal {I}}_{p}^{2}}. The differential of a smooth function f at p, denoted d f p {\displaystyle \mathrm {d} f_{p}}, is p / I p 2 {\displaystyle _{p}/{\mathcal {I}}_{p}^{2}}.

A similar approach is to define differential equivalence of first order in terms of derivatives in an arbitrary coordinate patch.Then the differential of f at p is the set of all functions differentially equivalent to f − f (p) {\displaystyle f-f(p)} at p.
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