Use the divergence theorem to calculate the flux of the vector
field F⃗ (x,y,z)=−5xyi⃗ +2yzj⃗ +4xzk⃗ through...
Use the divergence theorem to calculate the flux of the vector
field F⃗ (x,y,z)=−5xyi⃗ +2yzj⃗ +4xzk⃗ through the sphere
S of radius 2 centered at the origin and oriented outward.
∬SF⃗ ⋅dA⃗ =
Use the extended divergence theorem to compute the total flux of
the vector field
F(x, y, z) = −3x2 + 3xz − y, 2y3 − 6y, 9x2 + 4z2 − 3x outward
from the region F that lies inside the sphere x2 + y2 + z2 = 25 and
outside the solid cylinder x2 + y2 = 4 with top at z = 1 and bottom
at z = −1.
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:
Verify the Divergence Theorem for the vector eld
F(x; y; z) = hy; x; z2i on the region E bounded by the planes y
+ z = 2,
z = 0 and the cylinder x2 + y2 = 1.
Surface Integral:
Triple Integral:
Find the flux of the vector field F =
x i +
e2x j +
z k through the surface S given
by that portion of the plane 2x + y +
8z = 7 in the first octant, oriented upward.
Use the Divergence Theorem to evaluate
S
F · dS,
where
F(x, y,
z) =
z2xi
+
y3
3
+ cos z
j +
(x2z +
y2)k
and S is the top half of the sphere
x2 +
y2 + z2 =
4.
(Hint: Note that S is not a closed surface.
First compute integrals over S1 and
S2, where S1 is the
disk
x2 +
y2 ≤ 4,
oriented downward, and S2 =
S1 ∪ S.)
Verify that the Divergence Theorem is true for the vector field
F on the region E. Give the flux. F(x, y, z) = xyi + yzj + zxk, E
is the solid cylinder x2 + y2 ≤ 144, 0 ≤ z ≤ 4.
Compute the line integral of the vector field F(x, y, z) = ⟨−y, x,
z⟩ along the curve which is given by the intersection of the
cylinder x 2 + y 2 = 4 and the plane x + y + z = 2 starting from
the point (2, 0, 0) and ending at the point (0, 2, 0) with the
counterclockwise orientation.
The vector field given by E
(x,y,z) = (yz – 2x)
x + xz y + xy
z may represent an electrostatic field?
Why? If so, finding the potential F a from which E may be
obtained.