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.

~ 169. 170.

Differentiation with respect to a parameter.

303

170. Substitution of new variables in the double integral.
The partition into superficial elements can be accomplished by
any arbitrary net of curves. Putting:
x = ((p, q), y-= {(p, q),
there is a curve belonging to
each constant value of p, and
likewise to each value of q.
9 '^I^ * Let us suppose each p-curve to
cut each q-curve in one point
/ 9 \7 within the domain of validity
of the double integral, and
let us consider the rectilinear
quadrilateral determined by the.o --------- points of intersection having the
Fig. 15. coordinates:
x - (p, q), x== - (p —+ 3, q), X3= p (p, q+4A), x4= - (p+Ap, q + Aq)
y1= P (p, ), y2 -A), (p- + 4 (+Ap, q, 3 ( qAq) Y4= (q +^Aq).
The area of this quadrilateral is:
r = - abs [(x4 - x) (y3 - y2) + (2 - x3) (y4 - y)].
The vanishing limit to which this expression tends when the quantities
Ap and A q converge to zero is at the same time the limit of the
quadrilateral bounded by the curves, and since:
x4 - x1 = 9pp -+ A, q + A ) - (p,) q) =- = dp+ d q
X2 - x3 = (P + Ap, q) -- (p, q +- A q)
- ~(P() + pb - i(paq) + 9 q - dp o - O — dq
4 - 1 (P + A, + Aq + - (2p q) - - dp + -p dqa
3 - s - (QP, q + A^ ) - (p + aP, ) - dq - a dp,| )
it is expressed by the differential:
dT= abs [dp q (f a av - adl *
a q )q Op /
Accordingly:
Jj )x dy (ff)x 4YJ f[(',) abs a- i f 1 dpdq.
lpl WY q 0p/
Employing polar coordinates:
*) The theorem of the total differential is applicable to the functions g9
and ip, when in each of the two systems the continuously variable direction
of the tangent depends also continuously on the points of the plane.