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

Sec. 9]_Line Integrals__277
expressed by the integral
A= ^Xdx + Ydy + Zdz.
c
If the force has a potential, i.e., if there exists a function U = U (x, y, z)
(a potential function or a force function) such that
dV dU dU
- = A ,
TT j , -7 L ,
dx dy dz
then the work, irrespective of the shape of the path C, is equal to
A =
(*. //2, *j) (*J, '/J. Z.)
Xdx + Ydy + Zdz^ dU = U(x 2t y tt
J
(*|. {/,, Zi) (*i,
J
l/ lt 2-J
z 2)-t/ (x lt y, z } ),
where (v l5 f/ 1? Zj) is the initial and (x 2 , 0).
s
x\
+ \
y\
= a
2294. \ r ,. --^= , where C is a segment of the straight line
2
A: |- if \- \
c K
connecting the points 0(0, 0) and A (I, 2).
2295.
^xyds,
where C is a quarter of the ellipse + = l
^i fi>
c
lying in the first quadrant.
2296. ifds, where C is the first arc of the cycloid x = a (t sin /)>
c
a (1 cos /).
2
2297. }x + y 2
ds, where C is an
c
circle x
arc of the involute of the
---- a (cos / (-/sin/), y = a(smt tcost) I0^/=^2ji].
2298.
^
(x 2
-\- y 2 2
)
c
d$, where C is an arc of the logarithmic spi-
ral r^ae m v(m>Q) from the point A (0, a) to the point 0( oo, 0).
2299. JU + y) rfs where C is the right-hand loop
c
niscate r 2 = a 2
cos2