Derivation

Warning: the definitions below are very similar

Definition

Derivation of $$C^\infty_p$$

A derivation at a point is any $\mathbb{R}-$Linear map:

$$D:C^\infty_p(\mathbb{R}^n)\rightarrow\mathbb{R}$$ that satisfies the Leibniz rule - that is $$D(fg)|_p=f(p)Dg|_p+g(p)Df|_p$$

Recall that

$$C^\infty_p(\mathbb{R}^n)$$ is a set of germs - specifically the set of all germs of smooth functions at a point

Derivation at a point

One doesn't need the concept of germs to define a derivation (at p), it can be done as follows:

$$D:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}^n$$ is a derivation if it is $\mathbb{R}-$Linear and satisfies the Leibniz rule, that is:

$$D(fg)=f(p)Dg + g(p)Df$$

Warnings

These notions are VERY similar (and are infact isomorphic (both isomorphic to the Tangent space)) - but one must still be careful.

See also

• Germ