136 Functions of Several Variables
of finitely many simple closed piece-wise smooth curves. We say that the
orientations of S and dS agree if S stays on your left as you walk
around the boundary in the direction of the orientation of the corresponding
curve with your head in the direction of the normal vector to S. If this is
possible, then S is orientable. An equivalent mathematical description is
as follows. Assume that the orientation of S is fixed and given by the unit
vectors ris that exist at most points of S. Then a small piece of S near dS
looks like a small piece of a domain in the (i, j) plane, with us defining
the direction of i x j. Then the positive orientation of dS means positive
orientation of the boundary of this two-dimensional domain as described
earlier.
A famous example of a smooth non-orientable surface is the Mobius
strip, first discovered by the German mathematician AUGUST FERDINAND
MOBIUS (1790-1868). You can easily make this surface by twisting a long
narrow strip of paper and taping the ends. Verify, for example, by drawing
a line along the strip, that the surface has only one side
We will now use integrals to provide coordinate-free definitions for the
divergence and curl of a vector field. A traditional presentation starts with
the definitions in the cartesian coordinates. Then, after proving Gauss's and
Stokes's Theorems, one can use the results to show that the definitions do
not actually depend on the coordinate system. We take a different approach
and start with the definitions that are not connected to any coordinate
system.
We begin with the DIVERGENCE. Divergence makes a scalar field out
of a vector field and is an example of an operator. This operator arises
naturally in fluid flow models. Let v = v(P) be the velocity at point P in
a moving fluid, and let p = p{P) be the density of the fluid at P. Consider
a small surface of area ACT with unit normal vector u. The rate of flow
per unit time, or flux, of fluid across ACT is pv • u ACT. For a region G
bounded by a smooth simple closed surface dG, the total flux across dG
is 3>(G) = JJ pv • uda. This flux is equal the rate of change in time of
G
the mass of fluid contained in G. The quantity $(G)/m(G), where m(G)
is the volume of G, is the average flux per unit volume in G; in the limit
m(G) —> G, that is, as the region G shrinks to a point, we get an important
local characteristic of the flow. This naturally leads to the definition of
divergence.
Let F be a continuous vector field with values in R^3. Then the