Question

Evaluate

S |

(9* x* +

S: z = x + y/2

, 0 ≤ x ≤ 4, 0 ≤ y
≤ 3

Answer #1

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.
S
(x + y + z) dS, S is the parallelogram with parametric
equations
x = u + v,
y = u − v,
z = 1 + 2u + v,
0 ≤ u ≤ 7,
0 ≤ v ≤ 4.

Evaluate ʃ ʃSF∙dS
where F = < x, y, 2z > over the surface of the cone z =
x2+y2 between z = 1
and z = 2 (downward orientation).

Evaluate ∫∫Sf(x,y,z)dS , where f(x,y,z)=0.4sqrt(x2+y2+z2)) and S
is the hemisphere x2+y2+z2=36,z≥0

Evaluate the flux, ∬SF⋅dS , of F(x,y,z)=yzj+z^2k through the
surface of the cylinder y^2+z^2=9 , z ≥ 0 , between the planes x=0
and x=3.

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 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.

Let S be the portion of the surface z=cos(y) with 0≤x≤4 and
-π≤y≤π. Find the flux of F=<e^-y,2z,xy> through S:
∫∫F*n 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

For f(x,y,z) = sqrt(35-x^2-4y^2-2z) 1. Find the gradient of
f(x,y,z) 2. Evaluate delta f(x,y,z) 3. Find the unit vectors U+ and
U- , that give the direction of steepest ascent and the steepest
descent respectively.

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