Andre W. answered • 01/22/14

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Let δn, δm, and δr be the uncertainties of n, m, and r, respectively. Then the error propagation law tells us that the relative uncertainty of the product F=4Π

^{2}n^{2}mr isδF/F = 2 δn/n + δm/m + δr/r,

from which you can find δF.

Note: a constant factor (4Π

^{2}) does not contribute to the relative uncertainty, whereas the*exponent*2 in n² becomes a*factor*of 2 in the relative uncertainty.Alternatively, you can compute the total differential of F,

dF = (∂F/∂n) dn + (∂F/∂m) dm + (∂F/∂r) dr

= (8Π

^{2}nmr) dn + (4Π^{2}n^{2}r) dm + (4Π^{2}n^{2}m) dr.
Andre W.

01/22/14