Question

Verify Divergence theorem for the function F=erhorho over the semi-cylindrical volume

V={x2+y2≤a
{0≤z≤h,x≥0

Answer 1

To verify the Divergence theorem for the function F = e_rho*rho over the semi-cylindrical volume V, evaluate the surface integral of F over the closed surface S that bounds V, and the triple integral of the divergence of F over V.

Step-by-step explanation:

To verify the Divergence theorem for the function F = e_rho*rho over the semi-cylindrical volume V = {x^2+y^2≤a, 0≤z≤h, x≥0}, we need to evaluate the surface integral of the vector field F over the closed surface S that bounds the volume V, and the triple integral of the divergence of F over the volume V.

Let's start by calculating the surface integral:

1. Calculate the outward unit normal vector to the surface S = {x^2+y^2=a, 0≤z≤h, x≥0}.
2. Use this normal vector and the vector field F to calculate the dot product F·dS.
3. Integrate this dot product over the surface S.

Next, let's calculate the triple integral:

1. Calculate the divergence of the vector field F.
2. Integrate this divergence over the volume V.

If the surface integral is equal to the triple integral, then the Divergence theorem is verified for the given function F over the specified volume V.