Proof: There is no function f ∈ C 2 ( R 3 ) with gradient ∇ f ( x , y , z ) = ( y z , x z , x y 2 )How can one show that there is no function, which is a continuously partially derivable function f ∈ C 2 ( R 3 ) with this gradient ∇ f ( x , y , z ) = ( y z , x z , x y 2 ).I thought about using the Hessian matrix since one has to calculate all second partial derivatives of f there.Since only the gradient is given, can I calculate the antiderivatives first: y z = x y z x z = x y z x y 2 = x y 2 zNow I want to calculate the antiderivatives of the antiderivatives: x y z = y z x 2 2 x y z = y z x 2 2 x y 2 z = y 2 z x 2 2 I didn't calculate the antiderivatives of the partial derivatives and I don't even know if that way is correct...

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Proof: There is no function   f  ∈      C    2    (            R      3        ) with gradient   ∇  f  (  x  ,  y  ,  z  )  =  (  y  z  ,  x  z  ,  x      y    2    )How can one show that there is no function, which is a continuously partially derivable function   f  ∈      C    2    (            R      3        ) with this gradient   ∇  f  (  x  ,  y  ,  z  )  =  (  y  z  ,  x  z  ,  x      y    2    ).I thought about using the Hessian matrix since one has to calculate all second partial derivatives of f there.Since only the gradient is given, can I calculate the antiderivatives first:  y  z  =  x  y  z  x  z  =  x  y  z  x      y    2    =  x      y    2    zNow I want to calculate the antiderivatives of the antiderivatives:  x  y  z  =            y      z              x        2              2    x  y  z  =            y      z              x        2              2    x      y    2    z  =                              y          2                z                  x          2                    2      I didn&#039;t calculate the antiderivatives of the partial derivatives and I don&#039;t even know if that way is correct...

svirajueh · Accepted Answer

Explanation:If   ∇  f  =  (      f    1    (  x  ,  y  ,  z  )  ,      f    2    (  x  ,  y  ,  z  )  ,      f    3    (  x  ,  y  ,  z  )  ) and       f    i  s are twice differentiable, we must have                              ∂          2                          f          1                            ∂        y        ∂        z              =                              ∂          2                          f          2                            ∂        x        ∂        z              =                              ∂          2                          f          3                            ∂        x        ∂        y            for all x, y, z but this doesn&#039;t hold here since   1  =  1  ≠  2  y    ∀  x  ,  y  ,  z  ∈      R  therefore such a function doesn&#039;t exist.

Proof: There is no function f ∈ C 2 </msup> ( <mrow c

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