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

Sec. 10] Surface Integrals 285
When we pass to the other side, S~, of the surface, this integral reverses
sign.
If the surface 5 is represented implicitly, F (x, y, z) = 0, then the direction
cosines of the normal of this surface are determined from the formulas
where
1 OF 1 Q dF 1 OF
COSa==-pr^-, COSB -FT^ COS V = -rr- -T , T ,
D dz
D dx
^ D dy
and the choice of sign before the radical should be brought into agreement
with the side of the surface S.
3. Stokes' formula. If the functions P = P (.v, //, z), Q = Q (x, //, z),
R = R(x, y, z) are continuously differentiable and C is a closed contour bounding
a two-sided surface S, we then have the Stokes' formula
(j)
C
rr\fdR dQ\ Id? dR\ a , fdQ dP\ 1 _
= \ \ 3-- 1
5- c s a + -3 T- cos fi + 3-2- T- 1 cos v l ^ dS,
JJ l\dy dz J ^\dt dx j
5
\dx dy J
where cos a, cos p, cosy are the direction cosines of the normal to the surface
S, and the direction of the normal is defined so that on the side of the
normal the contour S is traced counterclockwise (in a rigiit-handed coordinate
system).
Evaluate the following surface integrals of the first type:
2347.
$$ (* 8
4 tf)dS, where S is the sphere x z
+// 2
-{-z 2 = a*.
6
2348.
5$ Vx 2
-\-tfdS where 5 is the lateral surface of the
s
. s
cone + i==0 [O^z^bl
g_?6
Evaluate the following surface integrals of the second type:
2349. \ \ yz dydz -\-xzdz dx-\- xydxdy, where 5 is the external
s
side of the surface of a tetrahedron bounded by the planes A: 0,
y = Q t 2 = 0, x+y + z = a.
2350. Nzdxdy, where S is the external side of the ellipsoid
2351. x t
dydz-\-y*dzdx + z*dxdy, where S is the external
o
2
side of the surface of the hemisphere +// +? = a 2
(z ^0).
2352. Find the mass ot the surface of the cube O^x^l,
O^y^l, Os^z < 1, if the surface density at the point M (x, y, z)
Is equal to xyz.
y
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