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For Question 8 (as well as in general), I don't understand how to sketch velocity field arrows along the null clines as well as outside the null clines. For this question the f1 null cline would be when y-x=0 so y=x and the f2 null cline would be when y+x=0 so y=-x. I understand that along null cline 1 the velocity field arrows will be pointing upwards or downwards and along null cline 2 velocity field arrows will be pointing left or right but how do we determine whether they will be upwards or downwards or left or right? Also how would we determine the velocity field arrows outside the null clines?

Any help would be much appreciated.

user134785
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1 Answers1

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Well, even if you've asked to sketch vector field somewhere, you still can (and sometimes should or even must) do some calculations that will give you a rigorous understanding of vector field behaviour. You know the equations of null clines, so you could directly substitute coordinates of point from null cline to vector field and check signs of its components. For example, take the $f_1$ null cline: each point of it has coordinates $(x, x), \; x \in \mathbb{R}$. Then just plug $(x, x)$ into the $(y-x) {\bf e_1} + (y+x) {\bf e_2}$ and you'll obtain $2x \cdot {\bf e_2}$. So this gives you an exact formula which describes how vector field behaves along null cline $f_1$.

Now if you want to check vector field outside null clines, the easiest way to do this is (again!) to plug some point. Null clines define borders of domains where components have constant and definite sign. When you sketch null clines, the plane $\mathbb{R}^2$ will be split into several domains and you'll just have to check only one point from each domain. This would be enough to sketch direction of vector field everywhere.

Hope this will help.

Evgeny
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