Mathematics of Physics and Engineering

138 Functions of Several Variables

and, for a smooth simple closed planar curveC, the circulation ^ = §v-dr
c
measures the amount of vorticity. If S is the planar region enclosed by C,
then ^/m(S) is the average vorticity in the region, where m(5) is the area
of S. In the limit m(5) —> 0 we get another important local characteristic
of the flow. This naturally leads to the definition of the curl.
Let Fbea continuous vector field with values in R^3. Fix a point P and
a unit vector n and consider the plane passing through P and having n as
the normal vector. Let C be a piece-wise smooth simple closed curve in the
plane, and S, the part of the plane enclosed by the curve. We assume that
P G S, P £ C, and the orientation of C is counterclockwise as seen from
the top of n. By definition, the curl of F at the point P is the vector
curl F(P) so that, for all C and u,

cmlF(P).n= lim -L-JF-dr, (3.1.26)

m(S)—om(i>) Jc

where m(S) is the area of S. This definition assumes that the limit exists
for every unit vector fi, and, for each vector n, the limit does not depend
on the particular choice of the curves C.

EXERCISE 3.1.18? Let F = Fii+F 2 J+F 3 K and assume that the functions
Fi,F2, F3 have continuous first-order partial derivatives. Show that curl.F
can be computed as the vector given by the symbolic determinant

curLF

l J K

AAA dx dy dz

F\ F2 F3

(F3y-F2z)i+(Flz-F3x)3+(F2X-Fly)K. (3.1.27)

Hint: with P — (xo,yo,zo), take the plane through P parallel to one of the coor-
dinate planes and let C be the boundary of the square in that plane with center at
P and side a so that m(S) = a?. Taking the plane so that n = i, we find that

j> F -dr tta(F 3 {xo,yo + a/2,zo) - F 3 {x 0 ,yo -a/2,z 0 ))

+ a(F2(xQ,yo,zo -a/2) - F 2 (xQ,yo,z 0 + a/2));

draw the picture and keep track of the orientations. Dividing by a and passing
to the limit a —+ 0 gives the i component of curl F. The other two components
are computed similarly.
Relations (3.1.24) and (3.1.27) suggest the introduction of the vector