# Why is one term missing in the derivation of continuity equation in isentropic coordinates?

In the book by Jonathan E. Martin "Mid-Latitude Atmospheric Dynamics - A First Course"

the continuity equation for isentropic coordinates is derived by applying conservation of mass:

$$\frac{d}{dt}(\frac{\partial p}{\partial \theta}) + \frac{\partial p}{\partial \theta} \frac{\partial u}{\partial x}+ \frac{\partial p}{\partial \theta} \frac{\partial v}{\partial y}+\frac{\partial p}{\partial \theta} \frac{\partial}{\partial \theta}(\dot \theta)=0$$

Now, following the book, by expanding the total derivative on the left I would get:

$$\underbrace{\frac{\partial}{\partial t}(\frac{\partial p}{\partial \theta}) + u \frac{\partial }{\partial x}(\frac{\partial p}{\partial \theta}) + v \frac{\partial }{\partial y}(\frac{\partial p}{\partial \theta}) + \frac{d \theta}{dt}\cdot \frac{\partial }{\partial \theta}(\frac{\partial p}{\partial \theta})}_{\frac{d}{dt}(\frac{\partial p}{\partial \theta})} + \underbrace{\frac{\partial p}{\partial \theta} \frac{\partial u}{\partial x}+ \frac{\partial p}{\partial \theta} \frac{\partial v}{\partial y}}_{\frac{\partial p}{\partial \theta} \vec \nabla \cdot \vec V}+\frac{\partial p}{\partial \theta} \frac{\partial}{\partial \theta}(\dot \theta)=0$$

Using divergence operator, this has 6 terms, however, in the final result given by the book

I would miss the term

$$\frac{d \theta }{d t} \cdot \frac{\partial }{\partial \theta}(\frac{\partial p}{\partial \theta})$$

Why? I see no reason why this term is skipped.

Potential temperature is conserved considering adiabatic processes and thus: $$\text{d} \theta/\text{d}t = 0$$. Otherwise the term needs to be retained.