Use the Divergence Theorem to evaluate ∫∫ ? ∙ ?? where ? =
??2 ?⃑+ (??2...
Use the Divergence Theorem to evaluate ∫∫ ? ∙ ?? where ? =
??2 ?⃑+ (??2 − 3?4 )?⃑+
(?3 + ?2 )? and S is the surface of the
sphere of radius 4 with ? ≤ 0 and ? ≤ 0 . would spherical
coordinates work?
Use the Divergence Theorem to evaluate
S
F · dS,
where
F(x, y,
z) =
z2xi
+
y3
3
+ cos z
j +
(x2z +
y2)k
and S is the top half of the sphere
x2 +
y2 + z2 =
4.
(Hint: Note that S is not a closed surface.
First compute integrals over S1 and
S2, where S1 is the
disk
x2 +
y2 ≤ 4,
oriented downward, and S2 =
S1 ∪ S.)
Use the Divergence Theorem to calculate the surface integral S F
· dS; that is, calculate the flux of F across S. F(x, y, z) = x4i −
x3z2j + 4xy2zk, S is the surface of the solid bounded by the
cylinder x2 + y2 = 1 and the planes z = x + 8 and z = 0
Use the Stoke’s theorem to evaluate Z Z S (∇×F)·nˆ·dS where F(x,
y, z) = (x^2 z^2,y^2 z^2, xyz) and surface S is part of the
paraboloid z = x^2 + y^2 that lies inside the cylinder x^2 + y^2 =
4, oriented upwards. Sketch the surface S and label everything.
Use Stokes' Theorem to evaluate
C
F · dr
where C is oriented counterclockwise as viewed from
above.
F(x, y,
z) = (x +
y2)i +
(y +
z2)j +
(z +
x2)k,
C is the triangle with vertices
(7, 0, 0), (0, 7, 0), and (0, 0, 7).
Use Stokes' Theorem to evaluate C F · dr where C is oriented
counterclockwise as viewed from above. F(x, y, z) = xyi + 3zj +
7yk, C is the curve of intersection of the plane x + z = 10 and the
cylinder x2 + y2 = 9.
Use Stokes' Theorem to evaluate C F · dr where C is oriented
counterclockwise as viewed from above. F(x, y, z) = xyi + 3zj +
5yk, C is the curve of intersection of the plane x + z = 2 and the
cylinder x^2 + y^2 = 144.
Use the extended divergence theorem to compute the total flux of
the vector field
F(x, y, z) = −3x2 + 3xz − y, 2y3 − 6y, 9x2 + 4z2 − 3x outward
from the region F that lies inside the sphere x2 + y2 + z2 = 25 and
outside the solid cylinder x2 + y2 = 4 with top at z = 1 and bottom
at z = −1.