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De Rham derivative of a function

Contents

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^\infty(M) \to \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^\infty(M)^* \otimes \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.