Talk:Adjoint state method

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In the final section of the section "General case", I am pretty sure that there is a substantial error. It is said that
 * $$\langle \nabla j(v),\delta_v\rangle = d_v j(v;\delta_v) = d_v \mathcal{L}(u_v,v,\lambda_v;\delta_v). $$

The last equality seems to be wrong. From the second equation from the Lagrange multiplier ansatz we instead find
 * $$\langle \nabla j(v),\delta_v\rangle = d_v j(v;\delta_v) = d_vJ(u_v,v;\delta_v) = - \langle d_vD_v(u_v;\delta_v),\lambda_v\rangle. $$

The example in the linear case should be changed accordingly. To be specific, instead of
 * $$ \langle \nabla j(v),\delta_v\rangle = \langle Au_v,\delta_v\rangle + \langle \nabla_vB_v : \lambda_v\otimes u_v,\delta_v\rangle,$$

it should read
 * $$ \langle \nabla j(v),\delta_v\rangle = \langle Au_v,\delta_v\rangle = - \langle \nabla_vB_v : u_v\otimes \delta_v,\lambda_v\rangle.$$