Question

In the following problems, the surface S is the part of the paraboloid z= x^2 + y^2 which lies below the plane z= 4, and includes the circular intersection with this plane. This single surface S could also be described as being contained inside the cylinder x^2+y^2= 4.

(a) Iterate, but do not evaluate, the integral ∫∫S(z+x) dS in terms of two parameters. Write the integrand in simplest form.

(b) Use Stoke’s theorem to rewrite ∫S(delta X F) · ndS as a line integral, where F(x, y, z) = xzi + yzj + xyk. Then evaluate this integral.

Answer #1

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=

Given S is the surface of the paraboloid z= 4-x^2-y^2
and C is the curve of intersection of the paraboloid with the plane
z=0. Verify stokes theorem for the field F=2zi+xj+y^2k.( you might
verify it by checking both sides of the theorem)

Evaluate the surface integral (x+y+z)dS when S is part of the
half-cylinder x^2 +z^2=1, z≥0, that lies between the planes y=0 and
y=2

Evaluate the surface integral.
5. " S x 2 z dσ; S that part of the cylinder x 2 + z 2 = 1 which
lies between the planes y = 0 and y = 2, and is above the
xy-plane.

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.

Evaluate the surface integral.
S
z +
x2y
dS
S is the part of the cylinder
y2 +
z2 = 4
that lies between the planes
x = 0 and x = 3
in the first octant

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.

Evaluate the surface integral.
S
x2yz dS, S is the part of the plane
z = 1 + 2x + 3y
that lies above the rectangle
[0, 4] × [0, 2]

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 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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