An incompressible inviscid fluid, under the influence of gravity, has the velocity field $$\textbf u = (− \cos(x)\sin(y), \, \, \sin(x)\cos(y), \, \, 0)$$ with the $z$-axis vertically upwards, where $g$ is the acceleration due to gravity.
Use Euler’s equation to find the pressure $p$.
In my notes, the equation is given as $$ \frac { D \textbf u }{Dt} = \frac{\partial \textbf u }{\partial t} + (\textbf u \, \, . \triangledown ) \textbf u = - \frac1 {\rho} \triangledown p - g \textbf k$$ but what is $\textbf k$? I have also seen it written as $$ \frac { D \textbf u }{Dt} = \frac{\partial \textbf u }{\partial t} + (\textbf u \, \, . \triangledown ) \textbf u = - \frac1 {\rho} \triangledown p + \frac1{\rho} \textbf F $$ but what is $\textbf F$?
In our question, $\partial \textbf u / \partial t =0$ and I found $(\textbf u \, \, . \triangledown ) \textbf u$ is equal to: $$(- \sin (x) \cos (x) \sin ^2 (y) + \sin ^2 (x) \sin (y) \cos (y) , \sin (x) \cos (x) \sin ^2 (y) - \sin ^2 (x) \sin (y) \cos (y) , 0)$$
What do we do next?