## Problem on writing down a given vector field

Write down in $xy$-coordinates a vector field with the following plot.
• ## Solution

Recall that
Observing that all the vectors of the vector field are horizontal, we conclude that the $\mathbf{j}$ component of $\mathbf{F}$ is always 0.
Next, we try to understand how the $\mathbf{i}$ component behaves. We observe that positive values of $y$ correspond to rightward vectors and negative values of $y$ correspond to leftward vectors. We also observe that the length of the vectors increases as $y$ gets further from the origin. It appears that the $\mathbf{i}$ component is proportional to $y$.
This vector field may be given by $$\mathbf{F(x,y)} = y \ \mathbf{i}.$$