Curl of a Vector Field

For lamellar vector fields, the integral of the vector field around a closed path is zero. If the line integral of a vector field around a closed path is not zero, the field is known as non-lamellar vector field. Suppose a non-lamellar vector field is represented by several line of flow. Let us consider a plane rectangular area in this field. When the area is perpendicular to the field (position 1), the vector field is normal to each side of the rectangle. Hence the line integral along all side is zero. When the area is parallel to the field (position 2), the line integrals along BC and DA are zero, while line integrals along AB and CD have finite values. Thus, the line integral around the boundary of the area has a finite value since the value of the vector field along the upper and lower edges is assumed to be different. Thus the value of the line integral around a closed path depends upon the orientation of the small vector area considered in the region of the vector field. There is a certain orientation of the area, for which the line integral around a closed path depends upon the orientation of the small vector area considered in the region of the vector field. There is a certain orientation of the area, for which the line integral of the vector field is maximum.

The Curl of vector field at any point is defined as a vector quantity whose magnitude is equal to the maximum line integral per unit area along the boundary of an infinitesimal test area at that point and whose direction is perpendicular to the plane of the test area.

If A is any vector field at any point P and δS an infinitesimal test area at point P, then curl A at P is defined.

|curlsA|= Lim_(δS→0)⁡ [Φ_c A.dr ]_maximuum/δS〗                                 … (1)

Or curl A.n = Lim_(δS→0)⁡ [Φ_c A.dr]_maximuum/δS〗                                  … (2)

Here n is the unit vector along the normal to the plane of the test area.
Expression for Curl in Cartesian Coordinates

Curl v= ∇×v= |(i Ì‚&j Ì‚&k Ì‚ ∂/∂x & ∂/∂x & ∂/∂x V_x&V_y&V_z )|    
= i Ì‚(∂z/∂y- (∂v_y)/∂z)+ j Ì‚((∂v_x)/∂z- (∂v_z)/∂x)+ () Ì‚k((∂v_y)/∂x- (∂v_x)/∂y)
Notice that the curl of a vector function v is a vector.

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