Example 1. The motion of a fluid is described by the complex potential $f(z)=z^{2}$. Find the velocity potential, the stream function, the level lines, the streamlines, the magnitude and direction of the velocity vector $\mathbf{V}$, and the projections of the velocity vector $V_{O x}$ and $V_{o y}$ on the coordinate axes $O x$ and $O y$.
Solution. Assuming $z=x+i y$, we have

$$
f(z)=\left(x^{2}-y^{2}\right)+i 2 x y,
$$

from which the velocity potential $u(x, y)=x^{2}-y^{2}$ and the stream function $v(x, y)=2 x y$. The level lines $u(x, y)=$ const are hyperbolas $x^{2}-y^{2}=$ const. The streamlines $v(x, y)=$ const are hyperbolas $x y=$ const. The magnitude of the velocity vector

$$
V=\left|\overline{f^{\prime}(z)}\right|=|2 \bar{z}|=2 \sqrt{x^{2}+y^{2}} .
$$

The direction of the velocity vector is determined by the angle

$$
\varphi=-\arg f^{\prime}(z)=-\operatorname{arctg} \frac{y}{x}
$$

The projections of the velocity vector onto the coordinate axes $O x$ and $O y$

$$
V_{o x}=\frac{\partial u}{\partial x}=2 x, \quad V_{o y}=\frac{\partial u}{\partial y}=-2 y .
$$

The origin $O(0,0)$ is a stagnation point of the fluid.