Vector Fields              

 

 

Example of a vector field in a plane,   (x,y):             F   =   [ sin x] i  +  [ cos y] j     

 

 

Example of a vector field in space,   (x,y,z):     

 

     F   =   [sin x + exp(y)] i  +  [ln x – 3y + z2]j     +  [xyz] k    

    

 

 

 

 

 

 

Example of the Gradient  of a scalar field,    F  =  3x2  - 2y + 1/z

 

           Then        Grad F  =  ∂F/∂x i  +  ∂F/∂y j   +  ∂F/∂z k

 

      ∂F/∂x  =  6x,    ∂F/∂y  =  -2,   ∂F/∂z  =  -1/z2            Grad F  =  6x i  -  2 j   -  1/z2 k

 

 

 

 

 

Example of the Divergence of a Vector Field,  F        F  =  sin x  i   +  cos y  j   

 

        div F   =  ∂/∂x (sin x)  +  ∂/∂y (cos y)               div F  =    cos x   +  – sin y  

 

Note that this vector field is conservative.  i.e.  curl F  =  0

 

 

 

  

 

Example of the Curl of a Vector Field,  F           

 

             F   =    [sin x + exp(y)] i  +  [ln x – 3y + z2]j     +  [xyz] k

     

                                                      i                         j                        k

 

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

 

                                           sin x + exp(y)      ln x – 3y + z2            xyz

 

where  det is a  3x3  determinant

 

            curl F  =     [xz – 2z] i   - [yz – 0] j   + [1/x – exp(y)] k

 

 

 



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