De Rham derivative of a function
Definition with symbols
In other words, the de Rham derivative of a function sends a vector field to the function .
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:
However, the de Rham derivative is not tensorial. In other words, at a point cannot be determined from .
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.
Further information: Exact 1-form
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.