For the vector field F(r) = Ar3e-ar2rˆ+Br-3θ^ calculate the volume integral of the divergence over a sphere of radius R, centered at the origin.
Volume of sphere V= ∫∫∫dV = ∫∫∫r2sinθdrdθdφ
Force F(r) = Ar3e-ar2rˆ+Br-3θ^ where ^ denote basis (unit vectors), F is a vector and is a function of r which is also a vector.
D(V) divergence over volume
θ = zenith angle (limits 0 to π)
Φ = azimuthal angle (limits 0 to 2π)
R = radius, r = variable of integration (limits 0 to R)
The Attempt at a Solution
The divergence of this thing is A(3r2e-ar2-2ar4e-ar2)
Here is the integral I set up:
D(V) = A∫∫∫(3r2e-ar2-2ar4e-ar2)r2sinθdrdθdΦ and I got down to the following: ... = 12Aπ∫r4e-ar2dr - 8aAπ∫r6e-ar2dr. My problem is integrating this. I mean is this even possible? If this were complex I am sure I could do it... but the integral of e-ar2 alone is already difficult to integrate.e-r2 I know has the solution of √π when limits are from -∞ to ∞ but these limits are also different.