DLMF:1.5.E17 (Q140)

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DLMF:1.5.E17
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    Statements

    Defining formula
    ∇ 2 ⁡ f = ∂ 2 ⁡ f ∂ ⁡ x 2 + ∂ 2 ⁡ f ∂ ⁡ y 2 + ∂ 2 ⁡ f ∂ ⁡ z 2 = 1 ρ 2 ⁢ ∂ ∂ ⁡ ρ ⁡ ( ρ 2 ⁢ ∂ ⁡ f ∂ ⁡ ρ ) + 1 ρ 2 ⁢ sin 2 ⁡ θ ⁢ ∂ 2 ⁡ f ∂ ⁡ ϕ 2 + 1 ρ 2 ⁢ sin ⁡ θ ⁢ ∂ ∂ ⁡ θ ⁡ ( sin ⁡ θ ⁢ ∂ ⁡ f ∂ ⁡ θ ) . superscript ∇ 2 𝑓 partial-derivative 𝑓 𝑥 2 partial-derivative 𝑓 𝑦 2 partial-derivative 𝑓 𝑧 2 1 superscript 𝜌 2 partial-derivative 𝜌 superscript 𝜌 2 partial-derivative 𝑓 𝜌 1 superscript 𝜌 2 2 𝜃 partial-derivative 𝑓 italic-ϕ 2 1 superscript 𝜌 2 𝜃 partial-derivative 𝜃 𝜃 partial-derivative 𝑓 𝜃 {\displaystyle{\displaystyle\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2% }}+\frac{{\partial}^{2}f}{{\partial y}^{2}}+\frac{{\partial}^{2}f}{{\partial z% }^{2}}={\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^{2}\frac{% \partial f}{\partial\rho}\right)+\frac{1}{\rho^{2}{\sin^{2}}\theta}\frac{{% \partial}^{2}f}{{\partial\phi}^{2}}}+\frac{1}{\rho^{2}\sin\theta}\frac{% \partial}{\partial\theta}\left(\sin\theta\frac{\partial f}{\partial\theta}% \right).}}
    0 references
    DLMF id
    DLMF:1.5.E17
    0 references
    Symbols used
    partial derivative of $$f$$ with respect to $$x$$
    Defining formula
    ∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
    xml-id
    C1.S5.E3.m4ajdec
    0 references
    partial differential of $$x$$
    Defining formula
    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.m2akdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aedec
    0 references
    Q10760
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C1.S1.XMD1.m1cdec
    0 references
    Q10778
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C1.S5.XMD10.m1gdec
    0 references
    Q10780
    Defining formula
    ρ 𝜌 {\displaystyle{\displaystyle\rho}}
    xml-id
    C1.S5.XMD8.m1adec
    0 references
    Q10781
    Defining formula
    θ 𝜃 {\displaystyle{\displaystyle\theta}}
    xml-id
    C1.S5.XMD9.m1adec
    0 references
     
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