The potential flow theory in 2D is described from the incompressibility condition $${\displaystyle {\begin{aligned}{\nabla \cdot \mathbf u =0},&&{\mathbf u=\nabla \varphi}\,.\end{aligned}}}$$
$$\nabla^2\varphi=0$$
with the Cauchy-Riemann equations
$${\displaystyle {\begin{aligned}u&={\frac {\partial \varphi }{\partial x}}={\frac {\partial \psi }{\partial y}},&v&={\frac {\partial \varphi }{\partial y}}=-{\frac {\partial \psi }{\partial x}}\,.\end{aligned}}}$$
From there we can obtain practically all we need to model the flow.
We draw the velocity field using the vector potential $\psi$, since the vector potential is colinear with the velocity field.
Lamb and Batchelor define the stream function $\psi$ as follows: ${\displaystyle \psi (x,y,t)=\int _{A}^{P}\left(u\,\mathrm {d} y-v\,\mathrm {d} x\right)}$ . Where the velocity field is defined with :
$${\displaystyle \quad \mathbf {u} ={\begin{bmatrix}u(x,y,t)\\v(x,y,t)\\0\end{bmatrix}}.}$$
I heard that the stream-function $\psi$ is not defined for dimensions higher than 2.
Why is that?
How do we draw the velocity field in dimensions higher than 2 if $\psi$ is not defined and we need it to draw the velocity field, since it is colinear with the velocity vector?
