De Rham derivative of a function

Definition

Definition with symbols

Let $M$ be a differential manifold. The de Rham derivative is an operator that takes as input an infinitely differentiable function on $M$, and outputs a differential 1-form on $M$. It is defined as follows:

$df:=X\mapsto Xf$

In other words, the de Rham derivative of a function $f$ sends a vector field $X$ to the function $Xf$.

Facts

Where it lives

The de Rham derivative can be viewed as:

$d:C^{\mathrm{\infty }}(M)\to \mathrm{\Gamma }(T^{*}M)$

i.e. it is a map from the algebra of infinitely differentiable functions, to the space of differential 1-forms.

It can also be viewed as an element: $d\in C^{\mathrm{\infty }}(M{)}^{*}\otimes \mathrm{\Gamma }(T^{*}M)$

Not tensorial

However, the de Rham derivative is not tensorial. In other words, $df$ at a point $p\in M$ cannot be determined from $f(p)$.

Leibniz rule

Further information: de Rham derivative satisfies Leibniz rule

The de Rham derivative satisfies the Leibniz rule, namely:

$d(fg)=f(dg)+g(df)$

This is a direct consequence of the Leibniz rule for derivations.

Related notions

Exact 1-form

Further information: Exact 1-form

An exact 1-form is a differential 1-form, or a section of the cotangent bundle, that can be expressed as the de Rham derivative of a function.

de Rham derivative of a differential 1-form

Further information: de Rham derivative of a differential 1-form

There is a closely related notion called the de Rham derivative of a differential 1-form.