sinceSince the Jacobian is injective (full rank: <math>m</math>), a local (not necessarily unique) [[left inverse]], say <math>r^*_\mathbf x</math> with Jacobian <math>\mathbf R^*_\mathbf x</math>, exists such that <math>r^*_\mathbf x(e_\mathbf x(\tilde x))=\tilde x</math> and <math>\mathbf R^*_\mathbf x\mathbf{T_x}=\mathbf I_m</math>. In practice we do not need the left inverse function itself, but we ''do'' need its Jacobian, for which the above equation does not give a unique solution. We can however enforce a unique solution for the Jacobian by choosing the left inverse as, <math>r_\mathbf x:\R^n\to\R^m</math>:
