Let σ be the closed surface consisting of the portion of the paraboloid z = x 2 + y 2 for which 0 ≤ z ≤ 1 and capped by the disk x 2 + y 2 ≤ 1 in the plane z = 1. Find the flux of the vector field F ( x , y , z ) = z j − y k in the outward direction across σ .
Let σ be the closed surface consisting of the portion of the paraboloid z = x 2 + y 2 for which 0 ≤ z ≤ 1 and capped by the disk x 2 + y 2 ≤ 1 in the plane z = 1. Find the flux of the vector field F ( x , y , z ) = z j − y k in the outward direction across σ .
Let
σ
be the closed surface consisting of the portion of the paraboloid
z
=
x
2
+
y
2
for which
0
≤
z
≤
1
and capped by the disk
x
2
+
y
2
≤
1
in
the plane
z
=
1.
Find the flux of the vector field
F
(
x
,
y
,
z
)
=
z
j
−
y
k
in the outward direction across
σ
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Find the flux of the vector field V(x, y, z) = 9xy2i + 9x2yj + z3k out of the unit sphere.
Let S be the part of the plane 4x +2y+z=4 which lies in the first octant, oriented upward. Find the flux of the vector field F-3i+1j+2k across the surface S
Calculate the curl of the vector
=
(2x + 3y)i + (y+z)j + (xy + 2zx)k
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