Network Theory and Discrete Calculus – Differentiation Rules
This post is part of a series
In the previous post, we introduced discrete calculus on a binary tree. In particular, we introduced two sets of basis 1-forms we’ll refer to as
- Graph bases and
- Coordinate bases
and saw that discrete 1-forms can be expressed in either left- or right-components forms
where, in general, the left- and right-component 0-forms do not coincide, i.e.
due to the noncommutativity of discrete 0-forms and discrete 1-forms.
Recall the exterior derivative of a discrete 0-form may be expressed in left-component form as
Although the product rule is satisfied, i.e.
note the discrete 0-form is on the right of the discrete 1-form and the discrete 0-form is on the left of the discrete 1-form . Attempting to express the product rule in left components, we find
In order to move to the left of the coordinate bases above, we need to know the commutation relations
These commutation relations may be determined by noting that for any two discrete 0-forms and , we have
where use has been made of the coordinate commutation relations in the previous post.
Putting everything together, we find the product rule above implies the left components satisfy
Change of Variables
Change of variables is something straightforward, yet has many applications, so it is worth writing it down here for future reference.
Let be a discrete 0-form with
If is invertible, we can rewrite this as
Given any other discrete 0-form , we have
From this, we can read off the discrete chain rules