DLMF:30.13.E6 (Q8919)

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DLMF:30.13.E6
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    Statements

    test
    ∇ 2 = 1 h ξ ⁢ h η ⁢ h ϕ ⁢ ( ∂ ∂ ⁡ ξ ⁡ ( h η ⁢ h ϕ h ξ ⁢ ∂ ∂ ⁡ ξ ) + ∂ ∂ ⁡ η ⁡ ( h ξ ⁢ h ϕ h η ⁢ ∂ ∂ ⁡ η ) + ∂ ∂ ⁡ ϕ ⁡ ( h ξ ⁢ h η h ϕ ⁢ ∂ ∂ ⁡ ϕ ) ) = 1 c 2 ⁢ ( ξ 2 - η 2 ) ⁢ ( ∂ ∂ ⁡ ξ ⁡ ( ( ξ 2 - 1 ) ⁢ ∂ ∂ ⁡ ξ ) + ∂ ∂ ⁡ η ⁡ ( ( 1 - η 2 ) ⁢ ∂ ∂ ⁡ η ) + ξ 2 - η 2 ( ξ 2 - 1 ) ⁢ ( 1 - η 2 ) ⁢ ∂ 2 ∂ ⁡ ϕ 2 ) . superscript ∇ 2 1 subscript ℎ 𝜉 subscript ℎ 𝜂 subscript ℎ italic-ϕ partial-derivative 𝜉 subscript ℎ 𝜂 subscript ℎ italic-ϕ subscript ℎ 𝜉 partial-derivative 𝜉 partial-derivative 𝜂 subscript ℎ 𝜉 subscript ℎ italic-ϕ subscript ℎ 𝜂 partial-derivative 𝜂 partial-derivative italic-ϕ subscript ℎ 𝜉 subscript ℎ 𝜂 subscript ℎ italic-ϕ partial-derivative italic-ϕ 1 superscript 𝑐 2 superscript 𝜉 2 superscript 𝜂 2 partial-derivative 𝜉 superscript 𝜉 2 1 partial-derivative 𝜉 partial-derivative 𝜂 1 superscript 𝜂 2 partial-derivative 𝜂 superscript 𝜉 2 superscript 𝜂 2 superscript 𝜉 2 1 1 superscript 𝜂 2 partial-derivative italic-ϕ 2 {\displaystyle{\displaystyle\nabla^{2}=\frac{1}{h_{\xi}h_{\eta}h_{\phi}}\left(% \frac{\partial}{\partial\xi}\left(\frac{h_{\eta}h_{\phi}}{h_{\xi}}\frac{% \partial}{\partial\xi}\right)+\frac{\partial}{\partial\eta}\left(\frac{h_{\xi}% h_{\phi}}{h_{\eta}}\frac{\partial}{\partial\eta}\right)+\frac{\partial}{% \partial\phi}\left(\frac{h_{\xi}h_{\eta}}{h_{\phi}}\frac{\partial}{\partial% \phi}\right)\right)=\frac{1}{c^{2}(\xi^{2}-\eta^{2})}\left(\frac{\partial}{% \partial\xi}\left((\xi^{2}-1)\frac{\partial}{\partial\xi}\right)+\frac{% \partial}{\partial\eta}\left((1-\eta^{2})\frac{\partial}{\partial\eta}\right)+% \frac{\xi^{2}-\eta^{2}}{(\xi^{2}-1)(1-\eta^{2})}\frac{{\partial}^{2}}{{% \partial\phi}^{2}}\right).}}
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    DLMF id
    DLMF:30.13.E6
    0 references
    Symbols used
    partial derivative of $$f$$ with respect to $$x$$
    test
    ∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
    xml-id
    C1.S5.E3.m4acdec
    0 references
    partial differential of $$x$$
    test
    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.m2acdec
    0 references
    Q12436
    test
    ξ 𝜉 {\displaystyle{\displaystyle\xi}}
    xml-id
    C30.S13.XMD1.m1edec
    0 references
    Q12437
    test
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C30.S13.XMD2.m1edec
    0 references
    Q12440
    test
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C30.S13.XMD3.m1cdec
    0 references
    Q12438
    test
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C30.S13.XMD4.m1ddec
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    Q12439
    test
    h ξ , h η , h ϕ subscript ℎ 𝜉 subscript ℎ 𝜂 subscript ℎ italic-ϕ {\displaystyle{\displaystyle h_{\xi},h_{\eta},h_{\phi}}}
    xml-id
    C30.S13.XMD5.m1cdec
    0 references
     
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