Question

Verify the divergence theorem when S is the closed surface
formed by sectioning oﬀ the cylinder x2 + y2 = 1 with the planes z
= 0 and z = 1 and ⃗ F = xy⃗ j + yz⃗ k.

Answer #1

here i at first find the surface integral by using Gauss divergence theorems. But when i want to find surface integrale simply without using divergence theorem I see that the second integration that is double integration of yz with respect to dxdy can not possible as we can not substitute z from the following curve x^2+y^2=1, z=0 to z=1 .

So i think the given function F is wrong.

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) =
x4i −
x3z2j
+
4xy2zk,
S is the surface of the solid bounded by the cylinder
x2 +
y2 = 9
and the planes
z = x + 4 and
z = 0.

Use Divergence theorem to evaluate surface integral S F ·n dA
where S is the surface of the solid enclosed by the tetrahedron
formed by the coordinate planes x = 0, y = 0 and z = 0 and the
plane 2x + 2y + z = 6 and F = 2x i − x^2 j + (z − 2x + 2y) k.

Use the divergence theorem to find the outward flux ∫ ∫ S
F · n dS of the vector field F = cos(10y + 5z) i + 9 ln(x2 +
10z) j + 3z2 k, where S is the surface of the region bounded
within by the graphs of z = √ 25 − x2 − y2 , x2 + y2 = 7,
and z = 0. Please explain steps. Thank you :)

Use the divergence theorem to find the outward flux (F · n) dS S
of the given vector field F. F = y2i + xz3j + (z − 1)2k; D the
region bounded by the cylinder x2 + y2 = 25 and the planes z = 1, z
= 6

Verify the Divergence Theorem for the vector field F(x, y, z) =
< y, x , z^2 > on the region E bounded by the planes y + z =
2, z = 0 and the cylinder x^2 + y^2 = 1.
By Surface Integral:
By Triple Integral:

Problem Verify the Divergence
Theorem for the vector fifield
F(x, y, z) = <y, x,
z^2>on the region E bounded by the planes
y + z = 2,
z = 0 and the cylinder x^2 + y^2 =
1.
1.Surface Integral:
2.Triple Integral:

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.

Use the Divergence Theorem to find the outward flux of F=9y
i+5xy j−6z k
across the boundary of the region D: the region inside the
solid cylinder x2+y2≤4 between the plane z=0
and the paraboloid z=x2+y2
The outward flux of F=9y i+5xy
j−6z k across the boundry of
region D is____

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 =...

Verify that the Divergence Theorem is true for the vector field
F on the region E. Give the flux. F(x, y, z) = 5xi + xyj + 4xzk, E
is the cube bounded by the planes x = 0, x = 2, y = 0, y = 2, z =
0, and z = 2.

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