Question

compute the flux of the vector field F through the parameterized
surface S. F= zk and S is oriented upward and given, for 0 ≤ s ≤ 1,
0 ≤ t ≤ 1, by x = s + t, y = s – t, z = s^{2} +
t^{2}.

the answer should be 4/3.

Answer #1

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) = yi − xj + 4zk, S is the hemisphere x^2 +
y2^ + z^2 = 4, z ≥ 0, oriented downward

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) = xz i + x j + y k S is the hemisphere x2 +
y2 + z2 = 4, y ≥ 0, oriented in the direction of the positive
y-axis. Incorrect: Your answer is incorrect.

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) = yi − xj + 2zk,
S is the hemisphere
x2 + y2 + z2 = 4,
z ≥ 0,
oriented downward

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) = y i − x j + z2 k S is the helicoid (with
upward orientation) with vector equation r(u, v) = u cos v i + u
sin v j + v k, 0 ≤ u ≤ 5, 0...

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.

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) =
x2i +
y2j +
z2k
S is the boundary of the solid half-cylinder0 ≤
z ≤
16 − y2
, 0 ≤ x ≤ 5

Evaluate the surface integral
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) = x i + y j + 9 k
S is the boundary of the region enclosed by the
cylinder
x2 + z2 = 1
and the planes
y = 0 and x + y =...

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