DLMF:1.6.E38 (Q219)

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DLMF:1.6.E38
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    Statements

    Defining formula
    ∫ 𝐜 𝐅 ⋅ d 𝐬 = ∫ a b 𝐅 ⁢ ( 𝐜 ⁢ ( t ) ) ⋅ 𝐜 ′ ⁢ ( t ) ⁢ d t = ∫ a b ( F 1 ⁢ d x d t + F 2 ⁢ d y d t + F 3 ⁢ d z d t ) ⁢ d t = ∫ 𝐜 F 1 ⁢ d x + F 2 ⁢ d y + F 3 ⁢ d z . subscript 𝐜 ⋅ 𝐅 𝐬 subscript superscript 𝑏 𝑎 ⋅ 𝐅 𝐜 𝑡 superscript 𝐜 ′ 𝑡 𝑡 subscript superscript 𝑏 𝑎 subscript 𝐹 1 derivative 𝑥 𝑡 subscript 𝐹 2 derivative 𝑦 𝑡 subscript 𝐹 3 derivative 𝑧 𝑡 𝑡 subscript 𝐜 subscript 𝐹 1 𝑥 subscript 𝐹 2 𝑦 subscript 𝐹 3 𝑧 {\displaystyle{\displaystyle\int_{\mathbf{c}}\mathbf{F}\cdot\mathrm{d}\mathbf{% s}=\int^{b}_{a}\mathbf{F}(\mathbf{c}(t))\cdot\mathbf{c}^{\prime}(t)\mathrm{d}t% =\int^{b}_{a}\left(F_{1}\frac{\mathrm{d}x}{\mathrm{d}t}+F_{2}\frac{\mathrm{d}y% }{\mathrm{d}t}+F_{3}\frac{\mathrm{d}z}{\mathrm{d}t}\right)\mathrm{d}t=\int_{% \mathbf{c}}F_{1}\mathrm{d}x+F_{2}\mathrm{d}y+F_{3}\mathrm{d}z.}}
    0 references
    DLMF id
    DLMF:1.6.E38
    0 references
    Symbols used
    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2adec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q10796
    Defining formula
    F j subscript 𝐹 𝑗 {\displaystyle{\displaystyle F_{j}}}
    xml-id
    C1.S6.XMD7.m1dec
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