An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 174.

Green's Theorem.

313

namely those at which any entrance of a parallel to the axis of x
coincides with its exit.
Constructing at each point o the normal to the boundary curve
and denoting the angle its direction entering the domain makes with
the positive axis of ordinates by 1(p, 2, 9P3, p4, this angle being
always measured in the same direction of rotation from the positive
axis of ordinates to the negative axis of abscissae, then at the entrances
o0, 03 the angles p, and Tp3 are always in the third or fourth quadrant,
but at the exits o2, o4, the angles T2, T4 are in the first or second
quadrant. Therefore if ds denote the positive value of the element
of the arc of a boundary curve, we have:
dy -- dssinp1, dy -dssin T2, dy -— dssin 3, dy=-dssinc4.
Accordingly:
f f(x,y) dxdy -=f (9 )sin p9ds + F(xl, y) sin (p1ls
+J F(X4, y) sin 94 ds +F(x3, y) sin (3 s.
All these partial integrals can be comprehended under the following
single conception:
We can describe each such integral as extended along a portion of
a boundary curve, inasmuch as the function that is to be integrated:
F(x, y) sin cp ds
has always to be formed for the continuously consecutive points of
the boundary curve, with positive values of ds.
Now we adopt as a convention: The length of the arc of any
boundary curve is a positive increasing magnitude, when we trace the
curve from any of its points so as to keep the bounded area on the
left. We thus obtain the partial integrals that the points 02 or o,
form along the segments: aca4, a2ca, a5a7, a7a,, aga1,, a11al3,
ac:al4 and from b1 by c1 to b2, b3 by C3 to b4; and again, those that
the points o0 or 03 form along the segments: a3al, a4a2, a6a3, casa,
a1oa8, a12 alo, a14a12 and from b2 by c2 to bh, b4 by c4 to b3. We can
therefore say: The integral
) F(x, y) sin p d s
is to be formed for the points of all the boundary curves,
these being traced so that the domain they have to bound
is constantly on the left.
We have conversely the definition: A simple integral
f F(x, y) sin p (s