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They suppose that the external force that it acts in each point of the fluid and the speed, derive from scalar , so they admit a potential :

[tex]\frac{\vec{F}}{m}= - \vec{\nabla} U[/tex]

[tex]\vec{V}= \vec{\nabla} \Omega[/tex]

If the acceleration depends on the coordinates of the point and the time [tex]\vec{a} ( u',v',w') = f(x,y,z,t)[/tex] :

[tex]u'= \frac{du}{dt}= \frac{\partial u}{\partial x} \frac{\partial x}{\partial t} +\frac{\partial u}{\partial y } \frac{\partial y}{\partial t} + \frac{\partial u}{\partial z}\frac{\partial z}{\partial t} + \frac{\partial u}{\partial t}[/tex] and thus with the other coordinates of the acceleration

And here my doubt comes, I do not understand as they obtain to this expression:

[tex]u'= \frac{\partial^2 \Omega}{\partial x^2} \frac{\partial \Omega}{\partial x} +\frac{\partial^2 \Omega}{\partial x \partial y} \frac{\partial \Omega}{\partial y} + \frac{\partial^2 \Omega}{\partial x \partial z}\frac{\partial \Omega}{\partial z} + \frac{\partial^2 \Omega}{\partial x \partial t}[/tex]

if somebody can help to understand it me, would be thanked for. Thank you very much