Part 3: Divergence and Curl

There are two types of "derivatives" of vector fields that occur frequently in applications.  To begin with, if F  = á M, N, P ñ, then the divergence of F( x,y,z) is defined to be

div  F =  Mx + Ny + Pz

Notice that the divergence of a vector field is a function and not a vector field. 

Similarly, the curl of F( x,y,z) is defined to be the new vector field

curl  F = áPy-Nz, Mz-Px, Nx-My ñ

Equivalently, the curl of a vector field can be defined formally by

curl  F  =  Ñ×F =   ¶ ¶x ,  ¶ ¶y ,  ¶ ¶z × á M,N,P ñ

Indeed, the formal cross product reduces to

Ñ×F = ê ê ê ê ê  ¶ ¶y  ¶ ¶z N P ê ê ê ê ê , ê ê ê ê ê  ¶ ¶z  ¶ ¶x P M ê ê ê ê ê , ê ê ê ê ê  ¶ ¶x  ¶ ¶y M N ê ê ê ê ê =  ¶P ¶y -  ¶N ¶z ,    ¶M ¶z -  ¶P ¶x ,    ¶N ¶x -  ¶M ¶y

which matches our original definition of the curl.

EXAMPLE 4    Calculate the divergence and curl of  F(x,y,z) = á x²y, xz, x²y³ ñ .      

Solution: Since M = x² y, N = xz, and P = x²y³, we have

div  F =     ¶ ¶x   x2y +   ¶ ¶y xz +  ¶ ¶z x2y3  =   2xy + 0 + 0  =   2xy

Similarly, the curl of F is given by

Ñ×F =  ¶ ¶y (x2y3) -  ¶ ¶z ( xz) ,  ¶ ¶z ( x2y) -  ¶ ¶x ( x2y3) ,  ¶ ¶x ( xz) -  ¶ ¶y ( x2y) = á 3x2y2-x,0-2xy3,z-x2 ñ = á 3x2y2-x,-2xy3,z-x2 ñ

       

Notice that if F = ÑU for some potential U, then F = á U_x,U_y,U_z ñ and

curl  F =    ¶Uz ¶y -  ¶Uy ¶z ,    ¶Ux ¶z -  ¶Uz ¶x ,    ¶Uy ¶x -  ¶Ux ¶y = á Uzy-Uyz,Uxz-Uzx,Uxy-Uyx ñ = á 0,0,0 ñ

The converse is true only over region R that is simply connected, where a simply-connected region is a connected region whose boundary is a simple closed curve.

Theorem 1: Let F(x,y,z) be differentiable in each component over an open, simply-connected region R. Then F is conservative over R if and only if

curl F = á 0,0,0 ñ

for all ( x,y,z) in R.

       

In section 3, we will develop methods for finding the potential of a conservative vector field.  For now, we will simply use the curl to determine if a vector field is conservative.

       

EXAMPLE 5    Determine if the vector field F(x,y,z) = á y,-x,z ñ is conservative       

Solution: To do so, we apply the formula for the curl of F with M = y, N = -x, and P = z:

curl  F = áPy-Nz, Mz-Px, Nx-My ñ =  ¶ ¶y ( -z) -  ¶ ¶z ( -x) ,  ¶ ¶z ( y) -  ¶ ¶x ( -z) ,  ¶ ¶x ( -x) -  ¶ ¶y ( y) = á 0,0,-2 ñ

Thus, curl  F ¹ 0 and F is not conservative.

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EXAMPLE 6    Determine if the following is conservative:

F( x,y,z) =   á yz, xz+2y, xy+1 ñ

Solution: To do so, we notice that M = yz, N = xz+2y, and P = xy+1, so that

curl  F =  ¶P ¶y -  ¶N ¶z ,  ¶M ¶z -  ¶P ¶x ,  ¶N ¶x -  ¶M ¶y =   á x-x,y-y,z-z ñ = á 0,0,0 ñ

Thus, F( x,y,z) =   áyz, xz+2y, xy+1 ñ is conservative.

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