DLMF:1.6.E22 (Q203)

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DLMF:1.6.E22
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    Statements

    test
    curl 𝐅 = ∇ × 𝐅 = | 𝐢 𝐣 𝐤 ∂ ∂ ⁡ x ∂ ∂ ⁡ y ∂ ∂ ⁡ z F 1 F 2 F 3 | = ( ∂ ⁡ F 3 ∂ ⁡ y - ∂ ⁡ F 2 ∂ ⁡ z ) ⁢ 𝐢 + ( ∂ ⁡ F 1 ∂ ⁡ z - ∂ ⁡ F 3 ∂ ⁡ x ) ⁢ 𝐣 + ( ∂ ⁡ F 2 ∂ ⁡ x - ∂ ⁡ F 1 ∂ ⁡ y ) ⁢ 𝐤 . 𝐅 ∇ 𝐅 𝐢 𝐣 𝐤 partial-derivative 𝑥 partial-derivative 𝑦 partial-derivative 𝑧 subscript 𝐹 1 subscript 𝐹 2 subscript 𝐹 3 partial-derivative subscript 𝐹 3 𝑦 partial-derivative subscript 𝐹 2 𝑧 𝐢 partial-derivative subscript 𝐹 1 𝑧 partial-derivative subscript 𝐹 3 𝑥 𝐣 partial-derivative subscript 𝐹 2 𝑥 partial-derivative subscript 𝐹 1 𝑦 𝐤 {\displaystyle{\displaystyle\operatorname{curl}\mathbf{F}=\nabla\times\mathbf{% F}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ \displaystyle{\frac{\partial}{\partial x}}&\displaystyle{\frac{\partial}{% \partial y}}&\displaystyle{\frac{\partial}{\partial z}}\\ F_{1}&F_{2}&F_{3}\end{vmatrix}\\ =\left(\frac{\partial F_{3}}{\partial y}-\frac{\partial F_{2}}{\partial z}% \right)\mathbf{i}+\left(\frac{\partial F_{1}}{\partial z}-\frac{\partial F_{3}% }{\partial x}\right)\mathbf{j}+\left(\frac{\partial F_{2}}{\partial x}-\frac{% \partial F_{1}}{\partial y}\right)\mathbf{k}.}}
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    DLMF id
    DLMF:1.6.E22
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    Symbols used
    Q10765
    test
    det {\displaystyle{\displaystyle\det}}
    xml-id
    C1.S3.SS1.m1aadec
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    partial derivative of $$f$$ with respect to $$x$$
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    ∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
    xml-id
    C1.S5.E3.m4acdec
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    partial differential of $$x$$
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    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.m2acdec
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    Q10760
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C1.S1.XMD1.m1cdec
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    Q10791
    test
    𝐢 𝐢 {\displaystyle{\displaystyle\mathbf{i}}}
    xml-id
    C1.S6.XMD2.m1gdec
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    Q10792
    test
    𝐣 𝐣 {\displaystyle{\displaystyle\mathbf{j}}}
    xml-id
    C1.S6.XMD3.m1gdec
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    Q10793
    test
    𝐤 𝐤 {\displaystyle{\displaystyle\mathbf{k}}}
    xml-id
    C1.S6.XMD4.m1gdec
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    Q10795
    test
    F j subscript 𝐹 𝑗 {\displaystyle{\displaystyle F_{j}}}
    xml-id
    C1.S6.XMD6.m1adec
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