Mathematics of Physics and Engineering

Integration and Differentiation 137

divergence of F at the point P is the number div F(P) so that

lim

m(G)-»o m(G)

div F(P) = lim _ —^— /"/" F • d<r, (3.1.23)

9G

where G = G(P) is a measurable solid region that contains the point P,
and m(G) is the volume of G. This definition assumes that the limit exists
and does not depend on the particular choice of the regions G. Interpreting
F as the velocity field of fluid, the right-hand side of expression (3.1.23)
determines the density of sources (div F(P) > 0) or sinks (div F(P) < 0) at
the point P. For an electric field, sources and sinks correspond to positive
and negative charges, respectively. The vector field F is called solenoidal
in a domain G if div F = 0 everywhere in G. The reason for this terminology
is that, in the theory of magnetism, there are no analogs of positive and
negative electric charges. Therefore, a magnetic field has no sources or sinks
and its divergence is zero everywhere. On the other hand, a magnetic field
is often produced by a solenoid — a coil of wire connected to an electric
source.

EXERCISE 3.1.17? Let F = F\ 1 + F2J + F3 k and assume that the functions
Fi,F2,Fs have continuous first-order partial derivatives. Show that

divF = Flx + F2y + F3z, (3.1.24)

where, as usual, F\x = dF\jdx, etc. Hint: let P = (xo,yo,zo) and consider a
cube G with center at P, faces parallel to the coordinate planes, and side a. Then
m(G) = a^3. The total flux through the two faces that are parallel to the (j, k)
plane is then approximately

(F 1 (x 0 + a/2,y 0 ,zo) - Fi(x 0 - a/2,y 0 ,z 0 ))a^2 ; (3.1.25)

draw a picture and keep in mind the orientation of the boundary. Then consider
the faces parallel to the other two coordinate planes. The total flux out of G is the
sum of the resulting three fluxes. After dividing by the volume of G and passing
to the limit, (3.1.25) results in Fix(xo,yo, zo), and Fiv,Fzz come form the fluxes
across the other two pairs of faces.
Next, we discuss the CURL. AS a motivation, we again consider a vector
field v representing the velocity of moving fluid or gas. The flow of fluid
or gas can have vortices: think of the juice in a blender or hurricanes,
tornadoes or large-scale circulation of atmospheric winds due to Coriolis
force. Vortices are caused by circular motion of the particles in the flow,