Visualizing the flow of a constant vector field: shouldn't all the flow lines (integral curves) be in the same direction as the vector field?

Question

A question in my text proposes the vector field $\vec v = x\vec j $. It shows that this:

as being the field itself; this makes sense. It also shows $\vec v $ 's flow as being this:

Why is it that all the flow lines are in the same direction? Shouldn't the ones in quadrants $2$ and $3$, where $x$ is negative, be pointing/flowing downwards?

Answer 1

I agree with you. The flow line through $(x_0,y_0)$ has equations $x(t) = x_0$, $y(t) = y_0 + x_0t$.

Answer 2

The "solution" to this equation is, for each point $(a,b)$, a curve $\gamma_{ab}$ passing through $(a, b)$ at some time (I'll use $t = 0$) , and with $$ \gamma_{ab}'(t) = v(\gamma(a,b)) $$

So here's the solution:

$$ \gamma_{ab}(t) = (a, b) + t (a,0) = ((1+t)a, b) $$ Clearly $\gamma_{ab}(0) = (a, b)$, so it passes through the point.

What's $\gamma_{ab}'(t)$? It's just $(a, 0)$, which is exactly the vector field at location $(a, (1+t)b)$