Question

use stoke's theorem to find ∬ (curl F) * dS where F (x,y,z) = <y, 2x, x+y+z> and and S is the upper half of the sphere x^2 + y^2 +z^2 =1, oriented outward

Answer #1

Use Stokes' Theorem to evaluate
S
curl F · dS.
F(x, y, z) = x2 sin(z)i + y2j + xyk,
S is the part of the paraboloid
z = 1 − x2 − y2
that lies above the xy-plane, oriented upward.

Use Stokes' theorem to find the flux curl ∫∫s (CurlG). dS where
G(x,y,z) = <-xy2, x2y, 1> and S is the
portion of the paraboloid z = x2 + y2 inside
the cylinder x2 + y2 = 1. Use an
upward-pointing normal.

Problem 10. Let F = <y, z − x, 0> and let S be the surface
z = 4 − x^2 − y^2 for z ≥ 0, oriented by outward-pointing normal
vectors.
a. Calculate curl(F).
b. Calculate Z Z S curl(F) · dS directly, i.e., evaluate it as a
surface integral.
c. Calculate Z Z S curl(F) · dS using Stokes’ Theorem, i.e.,
evaluate instead the line integral I ∂S F · ds.

Use the Divergence Theorem to evaluate ∫ ∫ S F ⋅ d S where
F=〈2x^3,2y^3,4z^3〉 and S is the sphere x2+y2+z2=16 oriented by
the outward normal.

Use Stokes" Theorem to evaluate (F-dr where F(x, y, z)=(-y , x-z
, 0) and the surface S is the part of the paraboloid : z = 4- x^2 -
y^2 that lies above the xy-plane. Assume C is oriented
counterclockwise when viewed from above.

Use Stokes' Theorem to compute the flux of
curl(F) through the
portion of the plane x37+y33+z=1 where
x, y, z≥0, oriented with an
upward-pointing normal, for F = <yz,
0, x>.
(Use symbolic notation
and fractions where needed.)
Flux =

Use Stokes’ Theorem to calculate the flux of the curl of the
vector field F = <y − z, z − x, x + z> across the surface S
in the direction of the outward unit normal where S : r(u, v)
=<u cos v, u sin v, 9 − u^2 >, 0 ≤ u ≤ 3, 0 ≤ v ≤ 2π. Draw a
picture of S.

Let F ( x , y , z ) =< e^z sin( y ) + 3x , e^x cos( z ) + 4y
, cos( x y ) + 5z >, and let S1 be the sphere x^2 + y^2 + z^2 =
4 oriented outwards Find the flux integral ∬ S1 (F) * dS. You may
with to use the Divergence Theorem.

Calculate the line integral of the vector field
?=〈?,?,?2+?2〉F=〈y,x,x2+y2〉 around the boundary curve, the curl of
the vector field, and the surface integral of the curl of the
vector field.
The surface S is the upper hemisphere
?2+?2+?2=36, ?≥0x2+y2+z2=36, z≥0
oriented with an upward‑pointing normal.
(Use symbolic notation and fractions where needed.)
∫?⋅??=∫CF⋅dr=
curl(?)=curl(F)=
∬curl(?)⋅??=∬Scurl(F)⋅dS=

F · dS
for the given vector field F and the oriented
surface S. In other words, find the flux of
F across S. For closed surfaces, use the
positive (outward) orientation.
F(x, y, z) = x2 i + y2 j + z2
k
S is the boundary of the solid half-cylinder 0 ≤ z
≤(9-y^2)^1/2
, 0 ≤ x ≤ 3
Please provide a final answer as this is where I have an
issue.

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