Scond Partial Derivative

(1) Definition (Second Partial Derivative) The higher-order partial derivatives for a function of two variables are denoted as


and


and the mixed second partial derivatives are denoted as


and

(2) Example (Second Partial Derivative) Find the second partial derivatives of .

    Solution. The first partial derivatives are

     and     

Therefore,

(3) Proposition (Clairaut's Theorem) If the function has mixed second-order partial derivatives and , that are continuous on an open disk containing , then

(4) Example (Second Partial Derivative) Verify that the function satisfies the wave equation

    Solution. We find that









and therefore,

(5) Example (Second Partial Derivative) Verify that the function is a solution of Laplace's equation

    Solution. We find that  









and therefore,

(6) Example (Second Partial Derivative) Verify that the functions and satisfy the Cauchy-Riemann equations
    

    
    Solution. We find that
    

and

(7) Example (Second Partial Derivative) Verify that the function is a solution of the three-dimensional Laplace equation  

    Solution. We compute



    

    

  Therefore,  

(8) Example (Second Partial Derivative) Show that the function is a solution of the equation

  

    Solution.  We compute

    

    

    

  and so
   
   
  as desired.

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