Example 10.5: Verify the divergence theorem for the vector field
F = 2xzi + yzj +z2k...
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
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.
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:
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 using an example that vector a + (vector b * vector c) is
not equal to (vector a + vector b) * (vector a + vector c) explain
the problem that arrises
Vector Analysis: Verify Green’s Theorem in the plane for ? ⃑ =
(?^2 + ?^2)?̂+ (?^2 − ?^2)?̂ in the anti-clockwise direction around
the ellipse 4?^2 + ?^2 = 16.
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 = 1 and the planes z = x + 8 and z = 0