Using Divergence theorum evaluate the flux integrate from s vector A.vector D's where A =xi^+yj^+zk^ and s is the complete surface of the hemisphere which has its centre at the origin and the origin and a radius of 6 units.
To evaluate the flux of the vector field A = xi + yj + zk across the complete surface (s) of the hemisphere, we can use the divergence theorem.
The divergence theorem states that the flux of a vector field across a closed surface is equal to the volume integral of the divergence of the vector field over the region enclosed by the surface.
In this case, the closed surface is the complete surface of the hemisphere of radius 6 units centered at the origin.
To calculate the flux, we need to follow these steps:
Step 1: Calculate the divergence of the vector field A.
The divergence of a vector field A = xi + yj + zk is given by the following formula:
div(A) = ∇ · A = ∂A/∂x + ∂A/∂y + ∂A/∂z
= ∂(xi)/∂x + ∂(yj)/∂y + ∂(zk)/∂z
= 1 + 1 + 1
= 3
Step 2: Calculate the volume integral of the divergence of A over the region enclosed by the hemisphere.
The region enclosed by the hemisphere can be defined as V = {(x, y, z) | x^2 + y^2 + z^2 ≤ 6^2, z ≥ 0}
Step 3: Apply the divergence theorem.
According to the divergence theorem, the flux (Φ) is given by the volume integral of the divergence of A over the region V:
Φ = ∭V div(A) dV
Step 4: Calculate the volume integral.
Since the vector field A is constant, the divergence is also constant. We can take it outside the integral:
Φ = div(A) * ∭V dV
Now, the volume integral of dV over the region V can be expressed as the volume of the hemisphere. The volume of a hemisphere is 2/3 * π * r^3, where r is the radius of the hemisphere. Plugging in the values, we have:
Φ = div(A) * (2/3 * π * 6^3)
Finally, substitute the value of div(A) and perform the calculation to obtain the flux (Φ).