Suppose I have 2 curves:
$$y=2 ± x^3$$
How would I find a 2D conservative vector field that is normal to both those curves?
I assumed that I had to parametrize each curve, but from then on I'm not sure how I would proceed.
2026-03-30 13:17:25.1774876645
Finding 2D conservative normal vector fields
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Do you know that the gradient of a function will always be perpendicular to its level curves? If you do then the answer becomes quick. Notice that the two curves are defined by
$$y-2\pm x^3 = 0$$
If we multiply the two together we get that
$$(y-2)^2-x^6 = 0$$
In other words $f(x,y) = (y-2)^2 - x^6$ has those curves defined as the level curves of $f(x,y) = 0$ so its gradient would satisfy the conditions.