**f**: R^{2}→R^{3} (where R is the real line) is defined by **f**(x, y) = (-y/(x^{2} + 4y^{2}), x/(x^{2} + 4y^{2}), 0). Can we find **F**: R^{3}→R^{3}, such that:

(1) if **F** = (F_{1}, F_{2}, F_{3}), then F_{i} all have continuous partial derivatives for all (x, y, z) ≠ (0, 0, 0);

(2) **∇ x F** = **0** for all (x, y, z) ≠ 0;

(3) **F**(x, y, 0) = **f**(x, y)?

**Solution**

Answer: no.

*Straightforward.*

We look first at (2). We know that **∇x F** = **0** iff **F** = **∇**g for some g: R^{3}→R. Remembering that tan^{-1}k has derivative 1/(1 + k^{2}) and playing around a little, we soon hit on g(x, y, z) = 1/2 tan^{-1}(2y/x) + h(z). That looks promising. In fact, it *almost* works. We might, for example take h(z) = 1/2 z^{2}, so that F(x, y, z) = (-y/(x^{2} + 4y^{2}), x/(x^{2} + 4y^{2}), z). That satisfies (2) and (3), and almost satisfies (1). The difficulty is that it, and its partial derivatives wrt x and y, are discontinuous (and arguably undefined) on the entire line x = 0, y = 0 (not just at the origin).

So we suspect that we *cannot* find F to satisfy all the conditions. The obvious approach is to assume we can and to seek a contradiction using Stokes' theorem ( ∫_{S} **∇x F .**d**a** = ∫_{Γ} **F.**d**s**, where S is a surface bounded by a closed curve Γ ). We want S to cut the z axis at z ≠ 0, because we believe the discontinuities are on the z-axis and so that will ensure that the surface integral is non-zero, whereas (2) would imply that it was zero. We want a curve Γ on which the line integral is easy to calculate. It clearly must lie entirely within the x, y plane (so that we can use (3) to give us F - otherwise we do not know what F is). It also looks sensible to take an ellipse x^{2} + 4y^{2} = k, because then f is drastically simplified. The line integral is then ∫_{Γ} **F.**d**s** = ∫_{Γ} **f.**d**s** = ∫_{Γ} (-y dx + x dy)/k = 2/k area ellipse ≠ 0. If (2) was satisfied, then the surface integral would be zero, since the surface avoids the origin. Contradiction.

© John Scholes

jscholes@kalva.demon.co.uk

7 Oct 1999