{ u = e x c o s ( y ) v = e x s i n ( y ) Determine the inverse functions, and compute the Jacobian of the inverse functions of these functions.

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{                            u          =                      e            x                    c          o          s          (          y          )                                      v          =                      e            x                    s          i          n          (          y          )                        Determine the inverse functions, and compute the Jacobian of the inverse functions of these functions.

antantil0 · Accepted Answer

Note that       u    2    +      v    2    =      e          2      x        →  x  (  u  ,  v  )  =            ln      ⁡      (              u        2            +              v        2            )        2  and       v    u    =  tan  ⁡  y  →  y  (  u  ,  v  )  =  arctan  ⁡      v    u  .Now, the Jacobian is       |                                        x            u                                                x            v                                                            y            u                                                y            v                                |    =      |                                        u                                          u                2                            +                              v                2                                                                          v                                          u                2                            +                              v                2                                                                                                    −              u                                                      u                2                            +                              v                2                                                                          v                                          u                2                            +                              v                2                                                          |    =            2      u      v              (              u        2            +              v        2                    )        2            (of course there are certain restrictions on domains and ranges from the arctan and the sines and cosines but whatever)

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