Verify the Divergence Theorem for the vector field F(x, y, z) =
< y, x ,...
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.
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:
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.
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.
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 =
Example 10.5: Verify the divergence theorem for the vector field
F = 2xzi + yzj +z2k and V is the volume enclosed by the upper
hemisphere x2 + y2 + z2 = a2, z ≥ 0
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.
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.)
Let f(x,y) be a scalar function, and let F(x,y,z) be a vector
field. Only one of the following expressions is meaningful. Which
one?
a) grad f x div F
b) div(curl(grad f))
c) div(div F)
d) curl(div(grad f))
e) grad(curl F)
Compute the derivative of the given vector field F. Evaluate the line integral of
F(x,y,z) = (y+z+yz , x+z+xz , x+y+xy )over the path C consisting of line segments joining (1,1,1) to (1,1,2), (1, 1, 2) to (1, 3, 2), and (1, 3, 2) to (4, 3, 2) in 3 different ways, along the given path, along the line from (1,1,1) to (4,3,2), and finally by finding an anti-derivative, f, for F.