## Archive for **January 2012**

## 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

related by

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.

and

due to the noncommutativity of discrete 0-forms and discrete 1-forms.

### Product Rule

Recall the exterior derivative of a discrete 0-form may be expressed in left-component form as

where

and

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

and

These commutation relations may be determined by noting that for any two discrete 0-forms and , we have

Therefore,

and

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

and

### 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

and