Question

. Let C be the curve x^{2}+y^{2}=1 lying in the
plane z = 1. Let ?=(?−?)?̂+?? =

(a) Calculate ∇×?

(b) Calculate ∫?∙?? F · ds using a parametrization of C and a chosen orientation for C.

(c) Write C = ∂S for a suitably chosen surface S and, applying Stokes’ theorem, verify your answer in (b)

(d) Consider the sphere with radius √22 and center the origin.
Let S’ be the part of the sphere that is above the curve (i.e.,
lies in the region z ≥ 1), and has C as boundary. Evaluate the
surface integral of ∇ × F over S′. Specify the orientation you are
using for S′.

Answer #1

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.

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=

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
=

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 − z j + y k
S is the part of the sphere
x2 + y2 + z2 = 4
in the first octant, with orientation toward the origin

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 − z j + y k
S is the part of the sphere
x2 + y2 + z2 = 25
in the first octant, with orientation toward the origin

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 - z j +
y k
S is the part of the sphere x2 +
y2 + z2 = 81 in the first
octant, with orientation toward the origin.

Let S be the boundary of the solid bounded by the paraboloid
z=x^2+y^2 and the plane z=16
S is the union of two surfaces. Let S1 be a portion of the plane
and S2 be a portion of the paraboloid so that S=S1∪S2
Evaluate the surface integral over S1
∬S1 z(x^2+y^2) dS=
Evaluate the surface integral over S2
∬S2 z(x^2+y^2) dS=
Therefore the surface integral over S is
∬S z(x^2+y^2) dS=

Calculate ∫ ∫S f(x,y,z)dS for the given surface and function.
x2+y2+z2=144, 6≤z≤12; f(x,y,z)=z2(x2+y2+z2)−1.

(1 point) Consider the paraboloid z=x2+y2. The plane 5x−3y+z−3=0
cuts the paraboloid, its intersection being a curve. Find "the
natural" parametrization of this curve. Hint: The curve which is
cut lies above a circle in the xy-plane which you should
parametrize as a function of the variable t so that the circle is
traversed counterclockwise exactly once as t goes from 0 to 2*pi,
and the paramterization starts at the point on the circle with
largest x coordinate. Using that...

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.

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