Global analysis. Exercises 8
1) Find M
 for
*  = (x - y)dx + (x + y)dy and M is the segment AB with A = (2, 3) and
B = (3, 5);
*  = ydx + xdy and M is a quarter of the circle g(t) = (R cos t, R sin t),
t  (0, 
2
);
*  = xdx + ydy + (x + y - 1)dz and M is the segment AB with A = (1, 1, 1)
and B = (2, 3, 4).
2) Find
M
dx3  dx4 + x1x3dx2  dx4,
where M = {(x1, x2, x3, x4)  R4
|x2
1 + x2
2 = 1, x2
3 + x2
4 = 1} and the orientation is
given by the parameterization g(u, v) = (cos u, sin u, cos v, sin v), (u, v)  (0, 2) ×
(0, 2).
3) Using the Stokes Theorem, find M
, where
*  = (x2
+ y2
)dx + (x2
- y2
)dy and M is the circuit of the triangle with the
vertexes A = (0, 0), B = (1, 0), C = (0, 1);
*  = (x2
+y2
)dx+(x2
-y2
)dy and M is the graph of the function y = 1-|1-x|,
x  (0, 2), starting at (0, 0);
*  = xydy  dz + yzdz  dx + xzdx  dy and M is the surface of the pyramid
bounded by the planes x = 0, y = 0, z = 0, x + y + z = 1.
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