Editing Differential (mathematics) (section)

=== Differentials as germs of functions ===

This approach works on any [[differentiable manifold]]. If
# {{var|U}} and {{var|V}} are open sets containing {{var|p}}
# <math>f\colon U\to \mathbb{R}</math> is continuous
# <math>g\colon V\to \mathbb{R}</math> is continuous
then {{var|f}} is equivalent to {{var|g}} at {{var|p}}, denoted <math>f \sim_p g</math>, if and only if
there is an open <math>W \subseteq U \cap V</math> containing {{var|p}} such that <math>f(x) = g(x)</math> for every {{var|x}} in {{var|W}}.
The germ of {{var|f}} at {{var|p}}, denoted <math>[f]_p</math>, is the set of all real continuous functions equivalent to {{var|f}} at {{var|p}}; if {{var|f}} is smooth at {{var|p}} then <math>[f]_p</math> is a smooth germ.
If
#<math>U_1</math>, <math>U_2</math> <math>V_1</math> and <math>V_2</math> are open sets containing {{var|p}}
#<math>f_1\colon U_1\to \mathbb{R}</math>, <math>f_2\colon U_2\to \mathbb{R}</math>, <math>g_1\colon V_1\to \mathbb{R}</math> and <math>g_2\colon V_2\to \mathbb{R}</math> are smooth functions
#<math>f_1 \sim_p g_1</math>
#<math>f_2 \sim_p g_2</math>
#{{var|r}} is a real number
then
#<math>r*f_1 \sim_p r*g_1</math>
#<math>f_1+f_2\colon U_1 \cap U_2\to \mathbb{R} \sim_p g_1+g_2\colon V_1 \cap V_2\to \mathbb{R}</math>
#<math>f_1*f_2\colon U_1 \cap U_2\to \mathbb{R} \sim_p g_1*g_2\colon V_1 \cap V_2\to \mathbb{R}</math>
This shows that the germs at p form an [[Algebra over a field|algebra]].

Define <math>\mathcal{I}_p</math> to be the set of all smooth germs vanishing at {{var|p}} and
<math>\mathcal{I}_p^2</math> to be the [[Ideal (ring theory)#Ideal operations|product]] of [[Ideal (ring theory)|ideals]] <math>\mathcal{I}_p \mathcal{I}_p</math>. Then a differential at {{var|p}} (cotangent vector at {{var|p}}) is an element of <math>\mathcal{I}_p/\mathcal{I}_p^2</math>. The differential of a smooth function {{var|f}} at {{var|p}}, denoted <math>\mathrm d f_p</math>, is <math>[f-f(p)]_p/\mathcal{I}_p^2</math>.

A similar approach is to define differential equivalence of first order in terms of derivatives in an arbitrary coordinate patch.
Then the differential of {{var|f}} at {{var|p}} is the set of all functions differentially equivalent to <math>f-f(p)</math> at {{var|p}}.