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.

~ 170. 171.

Substitution of new variables.

305

Third: the general case, when, retaining q, we introduce a
function p = (p', q) instead of p, and then employ the equation of
transformation obtained in the second case. Let us denote:
x = (Z) = (P', q), Y = — (Z, q) = q (PF, q),
thus corresponding to the second case we have:
0 (P (U]P) a q( P, q) (P) (p, qj U' dp'dq.
~f ( [a; P as q (IlP( q f (Oj( ) aO2 a4 ax
P-= (P, q)
Now since when p is regarded as a function of p' and q:
((p) = 4(p', q), ~(p, q) = V(p, q)
we have:
ll - 2 Up UW P U P
a _ a- azo ap a, a=
Up p U Up'' U ap 9
aq ap a aU q iq' a+ q q
therefore as a function of p' and q:
FU UWT U _ 1 __ U U ap a _p1 F rp Up U1
ap' )q p'i aq J L p p 'y W\ i
P=X (p', q)
accordingly the above double integral for x = (p', q), y = - (p', q):
f i(,y) x Ely =;( t abs a dp d ] q.
fXy~~)(dx, W abs p' Uq Up' U
171. When the function to be integrated becomes determinately
infinitely great or indeterminate between infinite limits at isolated
points in the domain, let us suppose each of these points surrounded
by a closed boundary curve. The question then arises: under what
condition is the function f(x, y) integrable in such a domain containing
a single infinity point? Let the coordinates of the point be x = a,
y = b and suppose it the centre of an arbitrarily small circle with
radius r^, the coordinates of any point upon or within this circle are:
x - a + r cos (p, y = b + r sin (,
where:
0<r<r,, 0<g<2r.
Now the necessary and sufficient condition that must be satisfied in
order that the double integral may present a determinate finite value
in this domain from its external boundary up to the circumference of
this circle, however small its radius r1 be taken, is: that the double
integral
Jf(a + r cos (p, b +- r sin cp)rdrdcp
extended over the interior of the circle with radius r1 must converge
to zero simultaneously with r,, or in other words, a superior limit
must be ascertainable for r1, such that the double integral extended
HARNACE, Calculus. 20