Question

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 :)

Answer #1

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

Use the Divergence Theorem to evaluate
S
F · N dS
and find the outward flux of F through the
surface of the solid bounded by the graphs of the equations.
F(x, y,
z) =
x2i +
xyj +
zk
Q: solid region bounded by the coordinate
planes and the plane 3x + 5y +
6z = 30

Use the Divergence Theorem to evaluate
S
F · N dS
and find the outward flux of F through the
surface of the solid bounded by the graphs of the equations.
F(x, y,
z) =
x2i +
xyj +
zk
Q: solid region bounded by the coordinate
planes and the plane 3x + 4y +
6z = 24

Use the Divergence Theorem to evaluate
F.N dS
and find the outward flux of F through the surface of the
solid bounded by the graphs of the equations.
F(x, y, z) = xi + xyj + zk
Q: solid region bounded by the coordinate planes and the plane
3x + 4y + z = 24

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 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____

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.

Evaluate the flux integral ∫ ∫ S F · n dS. F = 〈8, 0, z〉, S is
the boundary of the region bounded above by z = 25 − x2 − y2 and
below by z = 1 (n outward). Enter an exact answer. Do not use
decimal approximations.

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 across
the boundary of the region D.
F =x^2i -2xyj + 5xzk
D: The region cut from the first octant by the sphere
x^2+y^2+z^2=1

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