# De Rham derivative of a function

## Contents

## Definition

### Definition with symbols

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

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

## Facts

### Where it lives

The de Rham derivative can be viewed as:

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:

### Not tensorial

However, the de Rham derivative is *not* tensorial. In other words, at a point cannot be determined from .

### Leibniz rule

`Further information: de Rham derivative satisfies Leibniz rule`

The de Rham derivative satisfies the Leibniz rule, namely:

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.