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Given a scalar field $G$ on $\mathbb{R}^2$ (say), the vector field $(\frac{\partial G}{\partial x}, \frac{\partial G}{\partial y})$ is called the gradient of $G$.

Is there a standard name for the vector field $(-\frac{\partial G}{\partial y}, \frac{\partial G}{\partial x})$, which points along the contour lines of $G$ instead of orthogonally? (The relation between phase flow and the Hamiltonian, for example.)

Andrew Bacon
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    Don't you mean $(-\frac{\partial G}{\partial y}, \frac{\partial G}{\partial x})$ ? – Mark Fischler Jul 20 '15 at 21:35
  • Yes I did -- thanks for catching that. – Andrew Bacon Jul 20 '15 at 21:54
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    If one calls the former a gradient vector field, then it sounds natural to call the latter a tangent vector field (to the level sets of $G$). – A.Γ. Jul 20 '15 at 22:04
  • I like that terminology -- except I would say it is the tangent field (not the tangent vector field). For a function of three variables, the tangent field is in the space of 2-D surfaces. – Mark Fischler Jul 20 '15 at 22:23

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$(-\frac{\partial G}{\partial y}, \frac{\partial G}{\partial x})$ has a compact notation as $$ (z \times \nabla) G $$ But I don't know of any name for that expression. Note that $(z \times \nabla) G$ lies purely in the XY plane.

Perhaps there is no name for this expression is because the situation for fields on a 3-D space is such that there is no simple vector description of the contour surfaces, other than to tell what the normals to those surfaces are. And of course, the field of those normals is just the gradients.

Mark Fischler
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  • Thanks. Drawing on the Hamiltonian example above, perhaps you could call it "the flow of $G$". But the analogy only holds if we are in an even dimensional space where there's a natural way to pair off the dimensions. – Andrew Bacon Jul 21 '15 at 00:00