Distributional "Antiderivative"Suppose that R 3 , fix f ∈ L 1 ( R 3 ; R ) and let g ∈ L 1 ( R 3 ; R 3 ) satisfies d i v ( g ) = δ a − f ..Then what is g? It what is the distributional anti-derivative of this thing?

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Distributional &quot;Antiderivative&quot;Suppose that             R        3  , fix   f  ∈      L    1    (            R        3    ;      R    ) and let   g  ∈      L    1    (            R        3    ;            R        3    ) satisfies  d  i  v  (  g  )  =      δ    a    −  f  ..Then what is g? It what is the distributional anti-derivative of this thing?

komizmtk · Accepted Answer

Step 1Let             G      →        (            x      →        )  :=      1          4      π                          x        →                    ‖                        x          →                            ‖        3            . Then   div  ⁡            G      →        =      δ    0  . Now set                     g        →              p    :=            G      →        ∗  (      δ                            a          →                      −  f  ) i.e.                    g        →              p    (            x      →        )  =  ∫      1          4      π                                    x          →                    −                                    x            →                          ′                    ‖                        x          →                    −                                    x            →                          ′                    ‖        3              (      δ                            a          →                      (                    x        →              ′    )  −  f  (                    x        →              ′    )  )        d    3        x    ′      =      1          4      π                                    x          →                    −                        a          →                            ‖                        x          →                    −                        a          →                            ‖        3              −  ∫      1          4      π                                    x          →                    −                                    x            →                          ′                    ‖                        x          →                    −                                    x            →                          ′                    ‖        3              f  (                    x        →              ′    )        d    3        x    ′    .Step 2Then   div  ⁡                    g        →              p    =  div  ⁡  (            G      →        ∗  (      δ                            a          →                      −  f  )  )  =  (  div  ⁡            G      →        )  ∗  (      δ                            a          →                      −  f  )  =      δ    0    ∗  (      δ                            a          →                      −  f  )  =      δ                            a          →                      −  f  .But                     g        →              p   is not the only solution. We can to it add any                     g        →              h   such that   div  ⁡                    h        →              h    =  0.. All solutions are thus given by             g      →        =                    g        →              p    +                    g        →              h    .

tomekmusicd9 · Accepted Answer

Explanation:In dimension 3, it is classical that      d    i    v        (                            x          →                            ‖                              x            →                                    ‖          3                      )    =  4  π      δ    0  See for instance   ∇  ⋅            (                          r        ^                    r              2                        )       and Dirac Delta FunctionSo you can take   g  =      1          4      π            (                            x          →                            ‖                              x            →                                    ‖          3                      −                                        x            →                          −                              a            →                                      ‖                              x            →                          −                              a            →                                    ‖          3                      )

Distributional "Antiderivative" Suppose that <mrow class="MJX-TeXAtom-ORD"> <mi mathv

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