Integral of the substantial derivative in fluid mechanics? I am studying the book Modern Compressible Flow by John Anderson. He writes

For steady, inviscid, adiabatic, compressible flow with no body forces,
$$
\rho \frac{Dh_o}{Dt}=0
$$
where $h_o$ is the total enthalpy of the flow at a point.
When integrated, this yields $h_0=\text{const}$ along a streamline.

The substantial derivative is defined
$$
\frac{D}{Dt}=\frac{\partial }{\partial t} + u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y}+w\frac{\partial}{\partial z}
$$
My question is, how can I show that if
$$
\frac{Df}{Dt}=0
$$
for a function $f(x,y,z,t)$, if the flow is steady, adiabatic, and inviscid, then $f=\text{const}$ along a streamline?
 A: The substantial derivative (also known as the material derivative) is the rate of change of a variable $f$ in a fluid parcel or element. The variable $f$ is specified in the Eulerian way, as a function of time $t$ and current location $\mathbf{x}$ of the fluid parcel. Then the total derivative is
$$\mathrm{d}f\left(\mathrm{x}\left(t\right), t\right) = \frac{\partial f\left(\mathbf{x}\left(t\right), t\right)}{\partial t} \mathrm{d}t + \frac{\partial f\left(\mathbf{x}\left(t\right), t\right)}{\partial \mathbf{x}\left(t\right)} \frac{\partial \mathbf{x}\left(t\right)}{\partial t}\mathrm{d}t$$
where $\partial \mathbf{x}\left(t\right)/\partial t = \mathbf{u}\left(t\right)$ is just the flow velocity. Thus the material derivative is defined as
$$\frac{\mathrm{D}f\left(\mathrm{x}\left(t\right), t\right)}{\mathrm{D}t} \equiv \frac{\partial f\left(\mathbf{x}\left(t\right), t\right)}{\partial t} + \mathrm{grad}\left( f\left(\mathrm{x}\left(t\right), t\right)\right) \cdot \mathbf{u}\left(t\right).$$
If the material derivative, that is the total rate of change of the variable $f$, is zero, then the value of $f$ must be constant along its streamline. This is by definition. Or you could write it as
$$\frac{\mathrm{D}f}{\mathrm{D}t} =\frac{\partial f}{\partial t} + \mathrm{grad}\left( f\right) \cdot \mathbf{u}=0$$
thus
$$\mathrm{d}f = \left[\frac{\partial f}{\partial t} + \mathrm{grad}\left( f\right) \cdot \mathbf{u} \right]\mathrm{d}t = 0 \, \mathrm{d}t$$
and
$$f = \int \mathrm{d}f=\int 0 \, \mathrm{d}t = 0 + C$$
where $C$ is just the integration constant, i.e. the value of $f$ at the beginning of the streamline.
By the way: The conditions steady, adiabatic, and inviscid flow are not needed for this result. A variable $f$ does not change if $\frac{\mathrm{D}f}{\mathrm{D}t}=0$, but $\frac{\partial f}{\partial t} \neq 0$ and thus $\frac{\partial f}{\partial t} = - \mathrm{grad}\left( f\right) \cdot \mathbf{u}$ is still possible!
