Using the Divergence theorum evaluate the flux integration from s vector A.ds where A = xi^+yj^+zk^ and s is the complete surface of the hemisphere which has its centre at the origin and a radius of 6 units.
To evaluate the flux integration using the Divergence Theorem, we first need to compute the divergence of the vector A = xi + yj + zk.
The divergence of a vector field A = Ax i + Ay j + Az k is defined as the sum of the partial derivatives of its components with respect to their corresponding variables:
∇ · A = ∂Ax/∂x + ∂Ay/∂y + ∂Az/∂z
In this case, all the components of vector A = xi + yj + zk are linear functions with no dependency on the variables. So, the partial derivatives are constant, and their values are:
∂Ax/∂x = 1
∂Ay/∂y = 1
∂Az/∂z = 1
Therefore, the divergence of vector A is:
∇ · A = ∂Ax/∂x + ∂Ay/∂y + ∂Az/∂z = 1 + 1 + 1 = 3
Now that we have the divergence of the vector field A, we can use the Divergence Theorem to evaluate the flux integration over the surface of the hemisphere.
The Divergence Theorem states that the flux integral of a vector field A over a closed surface S equals the triple integral of the divergence of A over the volume V enclosed by S:
∬S A · dS = ∭V ∇ · A dV
In this case, we want to evaluate the flux integration over the complete surface of the hemisphere with center at the origin and radius 6 units. Since the hemisphere is a closed surface, we can apply the Divergence Theorem.
The volume enclosed by the hemisphere is half of the entire sphere with the same radius. The formula for the volume of a sphere with radius r is (4/3)πr^3. Therefore, the volume of the hemisphere with radius 6 units is:
V = (1/2) * (4/3) * π * 6^3 = 288π
Now we can evaluate the flux integration using the Divergence Theorem:
∬S A · dS = ∭V ∇ · A dV
∬S A · dS = ∭V (3) dV
∬S A · dS = 3 * V
∬S A · dS = 3 * 288π
∬S A · dS = 864π
Therefore, the flux integration of the vector A = xi + yj + zk over the complete surface of the hemisphere with radius 6 units is equal to 864π.