How can I show that ∫ Σ ( φ ψ , i − ψ φ , i ) d 2 x = ∫ Ω ( φ ψ , i i − ψ φ , i i ) d 3 x, where ψ , φ are scalar functions of the coordinates. Then i,ii designate differentiation of first and second order, and Σ , Ω are the surface and volyme respectively?

Question

How can I show that      ∫          Σ        (      φ              ψ              ,      i        −      ψ              φ              ,      i        )      d          2        x     =         ∫          Ω        (      φ              ψ              ,      i      i        −      ψ              φ              ,      i      i        )      d          3        x, where       ψ    ,      φ   are scalar functions of the coordinates. Then i,ii designate differentiation of first and second order, and       Σ    ,      Ω   are the surface and volyme respectively?

Kasen Schroeder · Accepted Answer

The Divergence theorem on             R      3       (Gauss Theorem) for the vector field       F    =  φ  ∇  ψ is                    (1)                              ∫          Σ                φ        ∇        ψ        ⋅                  d                Σ        =                  ∫          Ω                ∇        ⋅        (        φ        ∇        ψ        )                  d                Ω            With   ∇  ⋅  (  φ  ∇  ψ  )  =  ∇  φ  ⋅  ∇  ψ  +  φ      ∇    2    ψ and using             ∂      ψ              ∂      ν        =  ∇  ψ  ⋅            ν      ^       for the surface normal we can rewrite this as                    (2)                              ∫          Σ                φ                              ∂            ψ                                ∂            ν                                    d                Σ        =                  ∫          Ω                ∇        φ        ⋅        ∇        ψ                          d                Ω        +                  ∫          Ω                φ                  ∇          2                ψ                          d                Ω            This is a version of Greens first identityNow we swap the scalar functions   φ and   ψ in (2) and subtract from (2) and end up with                    (3)                              ∫          Σ                          (          φ                                    ∂              ψ                                      ∂              ν                                −          ψ                                    ∂              φ                                      ∂              ν                                )                          d                Σ        =                  ∫          Ω                          (          φ                      ∇            2                    ψ          −          ψ                      ∇            2                    φ          )                          d                Ω            This is a version of Greens second identity.

How can I show that int_sum (varphi psi_(,i)-psi varphi_(,i))d^2 x=int_Omega (varphi psi_(,ii)-psi varphi_(,ii))d^3 x , where psi, varphi are scalar functions of the coordinates.

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