lamellar field

A vector field $ \vec{F} = \vec{F}(x,\,y,\,z)$ , defined in an open set $ D$ of $ \mathbb{R}^3$ , is lamellar if the condition

$\displaystyle \nabla\!\times\!\vec{F} = \vec{0}$

is satisfied in every point $ (x,\,y,\,z)$ of $ D$ .

Here, $ \nabla\!\times\!\vec{F}$ is the curl or rotor of $ \vec{F}$ . The condition is equivalent with both of the following:

The scalar potential has the expression

$\displaystyle u = \int_{P_0}^P\vec{F}\cdot d\vec{s},$

where the point $ P_0$ may be chosen freely, $ P = (x,\,y,\,z)$ .

Note. In physics, $ u$ is in general replaced with $ V = -u$ . If the $ \vec{F}$ is interpreted as a force, then the potential $ V$ is equal to the work made by the force when its point of application is displaced from $ P_0$ to infinity.



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As of this snapshot date, this entry was owned by pahio.