Curl (or rotation), circulation, circulation intensity

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CURL (OR ROTATION), CIRCULATION, CIRCULATION INTENSITY

Def. Curl (or Rotation). Let A(x, y, z) = A₁ (x, y, z) i + A₂ (x, y, z) j + A₃ (x, y, z) k be a vector point function that is defined and differentiable at each point (x, y, z) in a certain region of space. The curl (or rotation) of A is a vector point function defined as

Curl. Physical interpretation.

Before giving the physical interpretation of the curl of a vector we define the concepts of “the circulation of a vector around a curve” and “the intensity of circulation at a point”.

Def. Circulation of vector A about curve C. Let A(x, y, z) = A₁ (x, y, z) i + A₂ (x, y, z) j + A₃ (x, y, z) k be a vector point function (i.e. vector field) that is defined and differentiable at each point (x, y, z) in a certain region G of space. Let C be a simple closed curve (i.e a closed curve that doesn’t intersect itself anywhere) within G defined by r(t) = x(t) i + y(t) j + z(t) k . Then the circulation of vector A about C is defined as

The integral

is a measure of the tendency of A to flow along the curve C.

Def. Intensity of circulation at point P. Pass a plane through a point P in region G (any plane). Let ΔS be a small segment of the plane which contains point P and which is bounded by the simple closed curve C. See Fig. 1. Then the intensity of circulation at point P is defined as

where the limit is taken in such away that ΔS shrinks to P. As one can see, dividing by ΔS yields a circulation per unit area of bounded surface, which is in the nature of an intensity of circulation.

In general, for each plane that passes through point P, there will be a different circulation intensity. For some plane passing through point P, there will be a maximum circulation intensity.

Physical interpretation of curl. Curl A at point P is that vector the direction of which is normal to the plane of the algebraically maximum circulation intensity at P and the magnitude of which is the maximum circulation intensity.

Example. If v is the velocity at a point P(x, y, z) in a moving fluid, the vector angular velocity of an infinitesimal portion of the fluid about P is given by ω = ½ curl v.

Proof

If a vector field F represents a moving fluid, for example, a paddle wheel placed at various points in the field would tend to rotate in regions where curl F ≠ 0. In areas where curl F = 0 there would be no rotation.

Def. Irrotational field. A field in which the curl is everywhere zero.

Def. Vortex field. A field that is not irrotational.