Vector Field

Curl Vector field
The vector field whose magnitude equals the maximum circulation at each point and to be oriented perpendicularly to this plane of circulation for each point is called the curl of a carrier field. It is denoted by curl(F) or  x F. Given the carrier field =P + Q + R the curl is defined as, curl (F) = (Ry- Qz) + (Pz- Rx) +(Qx- Py). Curl can be defined using the  operator given by =  + +. The same in the function form would be, =  + +.
The curl  =   x  which is the cross product given by, |                  |
|             |
|P       Q        R  |

Divergence of Vector field
Both curl and divergence are carrier operators. The dot product .  given by [, , , ……, ] is the divergence of the carrier field and here Fi ‘s are the component functions of the carrier field F.

A type of carrier field whose divergence is always zero is called Solenoidal Vector Field. If S(r) is a intended field then solenoidal is given by the dot product, . S which is always equal to zero.
Solenoidal carrier field being the only field whose divergence is zero and hence zero divergence is a way to find if the carrier field is solenoidal.  It is at times referred to as divergenceless field.
A Solenoidal carrier field is the only one that can be expressed as a curl of some other carrier, so we can write it as,  S(r) =  x A(r) where A(r) is the other carrier.

F is said to be conservative carrier field if there exists a scalar function f such that F = f where f is called the potential function for the carrier field. Conservative Vector Field Test would be, Given F a carrier field of class C1 the domain of which is simply connected region R in either R2  or R3, then F=f  for some scalar valued function f of class C2 on R if and only if  x F = 0, for all points of R.

Surface Integrals of Carrier fields is sometimes known as Flux of a Vector Field. Given a carrier field F with a normal carrier n then the surface integral F over the surface S would be  = , here the right hand integral is a standard surface integral. Sometimes this is called the flux of  across S.

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