MULTIPLE INTEGRALS
                   CHANGEofVARIABLES
     Change of Variables for Double Integrals
          • assume C1 transformations for (u,v) → (x,y)
                          x = g(u,v), y = h(u,v)
            for (u,v) ∈ S and (x,y) ∈ R;
          • define Jacobian for transformation, by determinant
                               ∂x ∂x 
                    ∂(x,y)     ∂u ∂v    ∂x∂y     ∂x∂y
                            =∂y ∂y  =         −        ;
                    ∂(u,v)     ∂u ∂v    ∂u∂v     ∂v∂u
          • transformed double integral formula is         
            ZZ               ZZ                     ∂(x,y)
                f(x,y)dA =        f(g(u,v),h(u,v))         dudv.
            R                 S                     ∂(u,v)
                     TRIPLE INTEGRALS
                   CHANGEofVARIABLES
    Change of Variables for Triple Integrals
         • assumeC1 transformationsfor(u,v,w) → (x,y,z)
             x = g(u,v,w), y = h(u,v,w), z = k(u,v,w)
           for (u,v,w) ∈ S and (x,y,z) ∈ R;
         • define Jacobian for transfomation, by determinant
                          ∂x ∂x ∂x 
                          ∂u ∂v ∂w 
            ∂(x,y,z)      ∂y ∂y ∂y 
                      =           
            ∂(u,v,w)      ∂u ∂v ∂w 
                          ∂z ∂z ∂z 
                          ∂u ∂v ∂w
                      =    ∂x∂y∂z + ∂z∂x∂y + ∂y∂z∂x
                           ∂u∂v∂w      ∂u∂v∂w     ∂u∂v∂w
                         −∂z∂y∂x −∂y∂x∂z −∂x∂z∂y;
                           ∂u∂v∂w     ∂u∂v∂w      ∂u∂v∂w
         • transformed triple integral formula is
       ZZZ f(x,y,z)dV
        R                                              
         ZZZ                                   ∂(x,y,z)
       =       f(g(u,v,w),h(u,v,w),k(u,v,w)            dudvdw.
          S                                   ∂(u,v,w)
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