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# Betti's theorem

Betti's theorem, which was discovered by Enrico Betti in 1872, states that for a linear elastic structure subject to two sets of forces {Pi} i=1,...,m and {Qj}, j=1,2,...,n, the work done by the set P through the displacements produced by the set Q is equal to the work done by the set Q through the displacements produced by the set P. This result is also known as the Maxwell-Betti reciprocal work (or reciprocity) theorem, and has applications in structural engineering where it is used to derive the Boundary element method.

$P \,\Delta_{Q1}=Q \,\Delta_{P2}$