Question

Find the flux of the vector field F (x, y, z) =< y, x, e^xz > outward from the z−axis and across the surface S, where S is the portion of x^2 + y^2 = 9 with x ≥ 0, y ≥ 0 and −3 ≤ z ≤ 3.

Answer #1

8. Use the Divergence Theorem to compute the net outward flux of
the field F= <-x, 3y, z> across the surface S, where S is the
surface of the paraboloid z= 4-x^2-y^2, for z ≥ 0, plus its base in
the xy-plane.
The net outward flux across the surface is ___.
9. Use the Divergence Theorem to compute the net outward flux of
the vector field F=r|r| = <x,y,z> √x^2 + y^2 + z^2 across the
boundary of the region...

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.

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.

Find the flux of the vector field F(x, y, z) = x, y, z through
the portion of the parabaloid z = 16 - x^2-y^2 above the
plane ? = 7 with upward pointing normal.

. Find the flux of the vector field F~ (x, y, z) =
<y,-x,z> over a surface with downward orientation, whose
parametric equation is given by r(s, t) = <2s, 2t, 5 − s 2 − t 2
> with s^2 + t^2 ≤ 1

Find the flux of the vector field F =
x i +
e6x j +
z k through the surface S given
by that portion of the plane 6x + y +
3z = 9 in the first octant, oriented upward.
PLEASE EXPLAIN STEPS. Thank you.

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.

Set up a double integral to find the flux of the vector field F
= <−x, −y, z^3 > through the surface S, where S is the part
of the cone z = sqrt( x^2 + y^2) between z = 1 and z = 3. You do
not have to evaluate the double integral.

Use the divergence theorem to calculate the flux of the vector
field F = (y +xz) i+ (y + yz) j - (2x + z^2) k upward through the
first octant part of the sphere x^2 + y^2 + z^2 = a^2.

Evaluate the outward flux ∫∫S(F·n)dS of the vector
fieldF=yz(x^2+y^2)i−xz(x^2+y^2)j+z^2(x^2+y^2)k, where S is the
surface of the region bounded by the hyperboloid x^2+y^2−z^2= 1,
and the planes z=−1 and z= 2.

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