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. 175.

Corollaries to Green's theorem.

315

the positive value of dx = ds sin 7p = ds cos 9p, but at their exits dx
is equal to - ds sin p =- - dscos; we prove the equation:
J jx!J) dyy = f — f(Fx, y) sin d s = JF(x, y) cos cpds.
By the above definition - F(x, y) cos Cp ds is the integral f (x, y) dx
formed along the boundary curves in a positive circuit.
175. Some consequences of this result claim our special attention.
1. Let a unique function of two variables f(x y) be given for a
domain limited in any manner by one or more boundary curves; let
its partial derived functions be:
a f(x,) f(xy) and let: -
W P, af~r~g)y, iandlet: } -
ax ay ay x
Now when the functions a u and a are integrable within the domain,
a(y ax
we have:
J dxdy =- fPcos (pds, - dxdy = Qsin pds,
therefore:
fJ f( -x aP) dxdy =j(Q sin (p + P cos (p) ds =f(Q dy + Pdx).
But by hypothesis the function under the double integral sign vanishes,
consequently we obtain the theorem:
If P and Q be the partial derived functions of a unique function
of two variables, the value of the integral:
J(P cos qp + Q sin ) ds == ( Pdx + Qdy)
is zero, when it is formed in a positive circuit for all the boundary
ap aQ
curves of a domain within which the functions y, a- are integrable
Gy ex
and in general, with possibly a linear set of exceptions, satisfy the
equation - aay ax
2. From this theorem follows: When the domain is limited only
by a single closed boundary curve (simply connected) and when
within it the conditions just stated are fulfilled, the value of the
integral formed for this one closed boundary curve is zero.
3. If two points within such a simply connected domain, whose
coordinates are xoyo, x1y1, be joined by arbitrary curves s1,s,, 3...
included within the domain, the value of the integral:
J(Pcos p + Q sin ) ds =f(Pdx - - Qdy)