Vector Fields   (continued)

 

 

Curl of a Vector Field in a plane, curl F,       where    F   =    F(x, y)

 

In general:  F(x, y)  =  P(x,y) i     +  Q(x,y) j   

 

  curl  F   =   (  ∂/∂x  i  +  ∂/∂y  j  ) x  ( P(x,y) i     +  Q(x,y) j  )     (cross product)

 

                                                                       

                          curl F  =    (  ∂Q/∂x   -  ∂P/∂y ) k

 

 

 

 

  

 

Curl of a Vector Field in space,  curl F ,        where    F   =    F(x, y, z)

 

In general:  F(x, y, z)  =  P(x,y,z) i     +  Q(x,y,z) j     +  R(x,y,z) k         

 

                                                            i             j              k

                          curl F  =     det      ∂/∂x       ∂/∂y         ∂/∂z

                                                           P            Q             R

 

where  det   means   determinant.

 

 

Expansion of this determinant gives for the curl of the vector field, F(x,y,z)

 

 

curl F  =   (  ∂R/∂y - ∂Q/∂z ) i  +  (  ∂P/∂z - ∂R/∂x) j  +  (  ∂Q/∂x - ∂P/∂y ) k   

 

 

 

 

Note:  A vector field may be conservative or non-conservative.  If the curl of the

vector field is zero, then the vector field is conservative.

 

A conservative vector field is said to be irrotational. 

A common application appears in the area of fluid mechanics.

 

 

 

Click here for examples.

 



Copyright © 2017 Richard C. Coddington

All rights reserved.