Gravitational Force: Formula Derivation for 2 Continuous Bodies

[demonelite123]

so the formula for gravitational force is F = Gm1m2/r² and that can be written as F = Gm1m2r^/r² (r^ represents a unit vector). now this formula is for 2 point masses. my teacher derived the formula for 2 continuous bodies using a 6 dimensional Riemann sum. he first considered the gravitational force between one small mass of one body (Δm1) and with every other small mass in the other body (Δm1', Δm2',...). so the sum of the forces between Δm1 and (Δm1', Δm2',...) is GΔm1Δm1'r^/r² + GΔm1Δm2'r^/r² + GΔm1Δm3'r^/r2...and when summed up and take limit n~> infinite, it becomes ∫GΔm1dm'r^/r² and he factored out the constants so it became GΔm1 ∫ r^/r² dm'. the limits of the integral are over the region of the body with small masses (Δm1', Δm2',...). then he took the integral again to sum up all these forces in over the region of the body with small masses (Δm1, Δm2,...) so he got

F = ∫G dm ∫r^/r² dm'. now I'm not sure what this means exactly. it just looks like 2 separate integrals. is this supposed to be a double integral? is this how you are supposed to write this equation? I've never seen integrals written this way before.

 

[ideasrule]

Would it be better if it was written as F=∫∫G dm*r^/r2 dm'? With double integrals, you can first integrate with respect to one variable, then integrate with respect to the other.

 

[demonelite123]

oh ok i wasn't sure if you could rearrange it into a double integral or not. thanks!