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Integral of vector product

  1. Mar 4, 2014 #1
    We know that: [tex]\frac{d}{dx}(\vec{f} \cdot \vec{g}) = \frac{d\vec{f}}{dx} \cdot \vec{g} + \vec{f} \cdot \frac{d\vec{g}}{dt}[/tex] and: [tex]\frac{d}{dx}(\vec{f} \times \vec{g}) = \frac{d\vec{f}}{dx} \times \vec{g} + \vec{f} \times \frac{d\vec{g}}{dt}[/tex] But, exist some formula (some expansion) for: [tex]\int \vec{f} \times \vec{g}\;\;dx[/tex] and for: [tex]\int \vec{f} \cdot \vec{g}\;\;dx[/tex] ?
     
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  3. Mar 4, 2014 #2

    HallsofIvy

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    The analogy with integration by parts would be
    [tex]\int\vec{f}\times\vec{g}dx= \vec{f}\times\int\vec{g}dx- \int \frac{d\vec{f}}{dx}\times \vec{g}dx[/tex]
    and
    [tex]\int\vec{f}\cdot\vec{g}dx= \vec{f}\cdot\int\vec{g}dx- \int \frac{d\vec{f}}{dx}\cdot \vec{g}dx[/tex]

    can you use your derivative formulas to verify those?
     
  4. Mar 4, 2014 #3

    vanhees71

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    I don't understand this conclusion. You get the integration-by-parts formula from integrating the differential expressions given in #1, e.g.,
    [tex]\frac{\mathrm{d}}{\mathrm{d} x} (\vec{f} \times \vec{g}) = \frac{\mathrm{d} \vec{f}}{\mathrm{d} x} \times \vec{g} + \vec{f} \times \frac{\mathrm{d} \vec{g}}{\mathrm{d} x}[/tex]
    wrt. [itex]x[/itex]
    [tex]\vec{f} \times \vec{g}=\int \mathrm{d} x \frac{\mathrm{d} \vec{f}}{\mathrm{d} x} \times \vec{g} + \int \mathrm{d} x \vec{f} \times \frac{\mathrm{d} \vec{g}}{\mathrm{d} x}[/tex]
    or bringing one term to the other side
    [tex]\int \mathrm{d} x \frac{\mathrm{d} \vec{f}}{\mathrm{d} x} \times \vec{g} = \vec{f} \times \vec{g} - \int \mathrm{d} x \vec{f} \times \frac{\mathrm{d} \vec{g}}{\mathrm{d} x}.[/tex]
     
    Last edited by a moderator: Mar 4, 2014
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