MATH.13:4.2 - Establish the problem-to-solution relation

Apply the transformation to the conditions. Show that an admitted solution is taken to an admitted solution of the stated target problem.

For an optimization problem, check feasibility and the criterion. If g maps the feasible set onto itself and J(g(x))=J(x), it maps every minimizer to a minimizer of that same problem.

For a rule f:X->Y, specify the input action g and the corresponding output action r(g). The relation

f(g(x))=r(g)(f(x))

is called equivariance: transforming the input and then computing agrees with computing and then transforming the output. Invariance is the case where the output stays unchanged. Classification of an object and prediction of its position can require these different relations.

For a dynamics law, transform the state, parameters and initial or boundary data. A transformed trajectory may solve the law with transformed data. Establish that correspondence before claiming anything about the solution with the original fixed data.

Use a mathematical argument for the range claimed. A few successful transformations can reveal or test a candidate symmetry; a conclusion covering an entire stated family requires the corresponding preservation argument.