Global analysis. Exercises 7
1) Check if the differential 1-form w is exact in the domain D. If it is exact, find all
functions  such that d = w:
* w = xydx + x2
2
dy, D = R2
;
* w = xdx + xzdy + xydz, D = R3
;
* w = ( 1
x2 + 1
y2 )(ydx - xdy), D = {(x, y)  R2
|xy = 0}.
2) Prove that the differential form w = r-3
(xdy  dz + ydz  dx + zdx  dy), where
r = x2 + y2 + z2, is closed in the domain R3
\{(0, 0, 0)}, but it is not exact there.
3) Find dw, g
w and g
(dw), where g : R2
 R3
, g(u, v) = (uv, u cosv, ev
):
* w = xdy
* w = ydz  dx
* w = dx  dy  dz
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