Verify Gauss divergence theorem of F = x i + y j + z k over the sphere x 2 + y 2 + z 2 = a 2

Question

Verify Gauss divergence theorem of   F  =  x  i  +  y  j  +  z  k over the sphere       x          2        +      y          2        +      z          2        =      a          2

Ashlynn Delacruz · Accepted Answer

Let       S    a   be your sphere, and       B    a   the enclosed ball. For each point       r    ∈      S    a   the outwards unit normal n is given by       n    =                    r            a      . Furthermore       F    (      r    )  =      r  . Since       r    ⋅      r    =      a    2   on       S    a   it follows that      ∫                  S        a                  F    ⋅      n          d    ω  =      ∫                  S        a                  r    ⋅                    r            a              d    ω  =  a      ∫                  S        a                  d    ω  =  4  π      a    3       .On the other hand,       d    i    v    (      F    )  ≡  3, and therefore      ∫                  B        a                  d    i    v    (      F    )        d    v    o    l    =  3        v    o    l    (      B    a    )  =  4  π      a    3       .

Verify Gauss divergence theorem of F=xi+yj+zk over the sphere x^2+y^2+z^2=a^2

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