Equation (1) may be written as
$$ \left({p}^{-1}\boldsymbol{\Omega} \right)\times \left(\rho \mathbf{u}\right)={\left(k-1\right)}^{-1}\mathbf{\nabla}\ln \sigma . $$
Applying the rotor operation to both parts of this equation, taking into account eq. (2) and equality div Ω = 0, valid since vorticity Ω = rot u, gives
$$ \rho \left(\mathbf{u}\cdot \nabla \right)\kern0.3em \left({p}^{-1}\boldsymbol{\Omega} \right)-\left(\kern0.2em \left({p}^{-1}\boldsymbol{\Omega} \right)\cdot \nabla \right)\left(\rho \mathbf{u}\right)-\rho \mathbf{u}\left(\kern0.3em \left({p}^{-1}\boldsymbol{\Omega} \right)\cdot \mathrm{\nabla ln}p\right)=0. $$
The flow velocity is represented in the form u = ue, ∣e ∣ = 1. Under the condition u ≠ 0, the last equation is equivalent to the following one
$$ \left(\mathbf{e}\cdot \mathbf{\nabla}\right)\left({p}^{-1}\boldsymbol{\Omega} \right)-\left(\left({p}^{-1}\boldsymbol{\Omega} \right)\cdot \mathbf{\nabla}\right)\mathbf{e}-\mathbf{e}\left(\left({p}^{-1}\boldsymbol{\Omega} \right)\cdot \mathbf{\nabla}\ln \left( p\rho u\right)\right)=0. $$
(3)
Let’s consider an arbitrary streamline with the vorticity Ω = Ω(l), where l is variable arc length along this streamline. Then we may write
$$ \left(\mathbf{e}\cdot \mathbf{\nabla}\right)\left({p}^{-1}\boldsymbol{\Omega} \right)=\frac{d}{dl}\left({p}^{-1}\boldsymbol{\Omega} \right). $$
(4)
Denote ex, ey, ez and Ωx, Ωy, Ωz unit vector e and vorticity vector Ω components in some Cartesian coordinate system Oxyz, F = ln(pρu). Using these notations and equality (4) it is possible to write vector eq. (3) along the streamline considered in the matrix form:
$$ \frac{d}{d\kern0.1em l}\left(\begin{array}{c}\frac{\Omega_x}{p}\\ {}\frac{\Omega_y}{p}\\ {}\frac{\Omega_z}{p}\end{array}\right)=\left(\begin{array}{ccc}\frac{\partial }{\partial x}{e}_x+{e}_x\frac{\partial }{\partial x}F& \frac{\partial }{\partial y}{e}_x+{e}_x\frac{\partial }{\partial y}F& \frac{\partial }{\partial z}{e}_x+{e}_x\frac{\partial }{\partial z}F\\ {}\frac{\partial }{\partial x}{e}_y+{e}_y\frac{\partial }{\partial x}F& \frac{\partial }{\partial y}{e}_y+{e}_y\frac{\partial }{\partial y}F& \frac{\partial }{\partial z}{e}_y+{e}_y\frac{\partial }{\partial z}F\\ {}\frac{\partial }{\partial x}{e}_z+{e}_z\frac{\partial }{\partial x}F& \frac{\partial }{\partial y}{e}_z+{e}_z\frac{\partial }{\partial y}F& \frac{\partial }{\partial z}{e}_z+{e}_z\frac{\partial }{\partial z}F\end{array}\right)\left(\begin{array}{c}\frac{\Omega_x}{p}\\ {}\frac{\Omega_y}{p}\\ {}\frac{\Omega_z}{p}\end{array}\right) $$
(5)
The matrix elements depend on both gasdynamic functions themselves and on their first derivatives, and are continuous (and limited) functions along arbitrary streamline, where velocity value u ≥ u0 > 0 with any arbitrary positive amount u0 > 0. If we consider the matrix elements to be given functions of the variable l, matrix eq. (5) represents the system of scalar ordinary differential equations for the vector p−1Ω components. This system is linear, its coefficients being continuous and limited. Hence, from the theorem of existence and uniqueness for systems of ordinary differential equations, it follows that along all the streamline considered either |p−1Ω| ≡ 0, or |p−1Ω| ≠ 0. Since the above reasoning is valid for an arbitrary value u0 > 0, we come to the following conclusion named vortex alternative:
If velocity value isn’t zero along some streamline of steady isoenergetic flow, then the vorticity value |Ω| either identically equals zero or |Ω| ≠ 0 along all the line.