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Curl of a Vector Field


The curl of a vector field A, denoted by curl A or ∇ x A, is a vector whose magnitude is the maximum net circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the net circulation maximum!.

In Cartesian


In Cylindrical



In Spherical



Given a vector field F(x, y, z) = Pi + Qj + Rk in space. The curl of F is the new vector field

This can be remembered by writing the curl as a "determinant"



Theorem: Let F be a three dimensional differentiable vector field with continuous partial derivatives.  Then Curl F = 0, if and only if F is conservative.

Example 1: Determine if the vector field F =  yz2i + (xz2 + 2) j + (2xyz - 1) k is conservative. 
Solution: 

Therefore the given vector field F is conservative.


Example 2: Find the curl of F(x, y, z) = 3x2i + 2zj – xk.
Solution:


Example 3: What is the curl of the vector field F = (x + y + z, x– z, x2 + y2 + z2)?
Solution:


Example 4: Find the curl of F = (x2 – y)i + 4zj + x2k.
Solution:



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