Question

Use Stokes' Theorem to evaluate

S |

curl **F** · d**S**.

F(x, y, z) = x^{2} sin(z)i + y^{2}j + xyk,

*S* is the part of the paraboloid

z = 1 − x^{2} − y^{2}

that lies above the *xy*-plane, oriented upward.

Answer #1

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.

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.

Evaluate the surface integral
S
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) = xy i + yz j + zx k
S is the part of the paraboloid
z = 4 − x2 − y2 that lies above the square
0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
and has...

Evaluate the surface integral
S
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) = xy i + yz j + zx k
S is the part of the paraboloid
z = 6 − x2 − y2 that lies above the square
0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
and has...

Evaluate the surface integral
S
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) = xy i + yz j + zx k
S is the part of the paraboloid
z = 6 − x2 − y2 that lies above the square
0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
and...

Evaluate the surface integral
S
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) = xy i + yz j + zx k
S is the part of the paraboloid
z = 2 − x2 − y2 that lies above the square
0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
and...

Let
F(x, y,
z) = z
tan−1(y2)i
+ z3
ln(x2 + 8)j +
zk.
Find the flux of F across S, the part
of the paraboloid
x2 +
y2 + z = 6
that lies above the plane
z = 5
and is oriented upward.
S
F · dS
=

Let F(x, y,
z) = z
tan−1(y2)i
+ z3
ln(x2 + 9)j +
zk. Find the flux of
F across S, the part of the paraboloid
x2 + y2 +
z = 7 that lies above the plane
z = 3 and is oriented upward.

Let F(x, y, z) = z tan−1(y2)i + z3 ln(x2 + 7)j + zk. Find the
flux of F across S, the part of the paraboloid x2 + y2 + z = 6 that
lies above the plane z = 5 and is oriented upward.

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) = ey
tan(z)i + y
3 − x2
j + x sin(y)k,
S is the surface of the solid that lies above the
xy-plane and below the surface
z = 2 − x4 − y4,
−1 ≤ x ≤ 1,
−1 ≤ y ≤ 1.

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