# Flow equation

Flow equation is a differential equation to describe how additive and multiplicative generators influence the assignment in an infinitesimal step.

## Formal definition

In the local polar coordinate system, it takes the form

 $\displaystyle{ \frac{da}{ds} = \mu \cos \theta + a \lambda \sin \theta }$ (1)

Or, in the local gradient-contour coordinate system, it takes the form

 $\displaystyle{ \frac{da}{ds} = \sqrt {\mu^2 + a^2 \lambda^2} \cos \phi }$ (2)

## The derivation of flow equation

Considering an infinitesimal generating process over a Riemannian manifold $\displaystyle{ M }$ with one generator for additional action $\displaystyle{ \mu }$ and the other for multiplicative action $\displaystyle{ e^\lambda }$, the two generators are perpendicular. Under this generation process, we get an assignment $\displaystyle{ a : M \to F }$ over the manifold.

At any point with an assignment $\displaystyle{ a_0 }$, considering a moving with distance of $\displaystyle{ \epsilon }$, direction angle of $\displaystyle{ \theta }$, time elapse of $\displaystyle{ \delta }$, we can establish

 $\displaystyle{ a_\delta = (a_0 + \mu \epsilon \cos \theta) e^{\lambda \epsilon \sin \theta} }$

or

 $\displaystyle{ a_\delta = a_0 e^{\lambda \epsilon \sin \theta} + \mu \epsilon \cos \theta }$

Both can be simplified to the same

 $\displaystyle{ a_\delta = a_0 + \epsilon (\mu \cos \theta + a_0 \lambda \sin \theta) }$

Then we have

 $\displaystyle{ \frac{1}{\delta}(a_\delta - a_0) = \frac{\epsilon}{\delta} (\mu \cos \theta + a_0 \lambda \sin \theta) }$

When both $\displaystyle{ \delta }$ and $\displaystyle{ \epsilon }$ are towards zero, we get $\displaystyle{ \frac{da}{dt} }$, and hence

 $\displaystyle{ \frac{da}{dt} = u (\mu \cos \theta + a \lambda \sin \theta) }$

Or, we can change it to other forms

 $\displaystyle{ \frac{da}{ds} = \mu \cos \theta + a \lambda \sin \theta }$