Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

282_Multiple and Line Integrals_[C/t. 7
2328. Applying Green's formula, evaluate
/ = 2 (x? -t- if) dx + (x + y) 2
where C is the contour of a triangle (traced in the positive direc-
tion) with verlices at the points A (I, 1), fl(2, 2) and C(l, 3).
Verify the result obiained by computing the integral directly.
2329. Applying Green's formula, evaluate the integral
c
x*y dx + xif dy,
where C is the circle x* + if = R* traced counterclockwise.
2330. A parabola AmB, whose axis is the #-axis and whose
chord is AnB, is drawn through the points A (1,0) and 8(2,3).
Find y (x + y)dx(x y)dy directly and by applying Green's
AmBnA
formula.
2331. Find
dy,
$ e*y [y* dx \- (1 -f xtj)dy\, if the points A and B
AmB
lie on the #-axis, while the area, bounded by the integration
path AmB and the segment AB, is equal to S.
2332*. Evaluate ^ifc^f. Consider two cases:
a) when the origin is outside the contour C,
b) when the contour encircles the origin n times.
2333**. Show that if C is a closed curve, then
where s is the arc length and n is the outer normal.
2334. Applying Green's formula, find the value of the integral
I = (j)[xcos(X, n)+ysm(X, n)]ds,
c
where ds is the differential of the arc and n is the outer normal to
the contour C.
2335*. Evaluate the integral
taken along the contour of a square with vertices at the points
A (1, 0). fl(0 f 1), C(-l, 0) and>(0, 1), provided the contour
is traced counterclockwise.