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:<math> \mathbf{v} = \nabla \varphi.</math>
The velocity potential is not uniquely defined since one can add to it an arbitrary function of time, say <math>f(t)</math>
In potential flow, the [[Circulation (physics)|circulation]] <math>\Gamma</math> around any [[Simply connected space|simply-connected contour]] <math>C</math> is zero. This can be shown using the [[Stokes theorem]],
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where {{math|∇<sup>2</sup> {{=}} ∇ ⋅ ∇}} is the [[Laplace operator]] (sometimes also written {{math|Δ}}). Since solutions of the Laplace equation are [[harmonic function]]s, every harmonic function represents a potential flow solution. As evident, in the incompressible case, the velocity field is determined completely from its [[kinematics]]: the assumptions of irrotationality and zero divergence of flow. [[Dynamics (physics)|Dynamics]] in connection with the momentum equations, only have to be applied afterwards, if one is interested in computing pressure field: for instance for flow around airfoils through the use of [[Bernoulli's principle]].
In incompressible flows, contrary to common misconception, the potential flow indeed
:<math>\mu\nabla^2\mathbf v = \mu\nabla(\nabla\cdot\mathbf v)-\mu\nabla\times\boldsymbol\omega=0</math>
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is identically zero. It is the inability of the potential flow to satisfy the required boundary conditions, especially near solid boundaries, makes it invalid in representing the required flow field. If the potential flow satisfies the necessary conditions, then it is the required solution of the incompressible Navier–Stokes equations.
In two dimensions,
==Compressible flow==
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:<math>\rho \nabla\cdot\mathbf v + \mathbf v\cdot\nabla \rho = 0, \quad (\mathbf v \cdot\nabla)\mathbf v= -\frac{1}{\rho}\nabla p = -\frac{c^2}{\rho}\nabla \rho</math>
where the last equation follows from
:<math>c^2\nabla\cdot\mathbf v - \mathbf v\cdot (\mathbf v \cdot \nabla)\mathbf v=0.</math>
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:<math>(c^2-\varphi_x^2)\varphi_{xx}+(c^2-\varphi_y^2)\varphi_{yy}-2\varphi_x\varphi_y\varphi_{xy}=0.</math>
'''Validity:''' As it stands, the equation is valid for any inviscid potential flows, irrespective of whether the flow is subsonic or supersonic (e.g. [[Prandtl–Meyer expansion fan|Prandtl–Meyer flow]]). However in supersonic and also in transonic flows, shock waves can occur which can introduce entropy and vorticity into the flow making the flow rotational. Nevertheless, there are two cases for which potential flow prevails even in the presence of shock waves, which are explained from the (not
:<math>\nabla (h+v^2/2) - \mathbf v\times\boldsymbol\omega = T \nabla s</math>
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:<math>2\alpha_*\frac{\partial\phi}{\partial x} \frac{\partial^2\phi}{\partial x^2} = \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2}</math>
where <math>\alpha_*</math> is the critical value of [[Landau derivative]] <math>\alpha = (c^4/
=== Unsteady flow ===<!-- [[Full potential equation]] redirects here -->
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