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Differentiation with respect vector

  1. Dec 10, 2013 #1

    For a long time I aks me if exist differentiation/integration with respect to vector and I think that today I discovered the answer! Given:
    So, df/dt is:
    [tex]\bigtriangledown f\cdot D\vec{r}[/tex]
    But, df/dt is:
    [tex]\frac{df}{d\vec{r}}\cdot \frac{d\vec{r}}{dt}[/tex]
    This means that:
    [tex]\frac{df}{d\vec{r}}=\bigtriangledown f[/tex]

    So, analogously, the integration with respect to vector is:
    [tex]\int f\;d\vec{r}=\bigtriangledown ^{-1}f[/tex]

    What make I think that:
    [tex]\frac{\mathrm{d} }{\mathrm{d} \vec{r}}=\bigtriangledown[/tex]

    Then, would be possible to formalize the differintegration of vector with respect to vector too by:
    [tex]\bigtriangledown \cdot \vec{f},\;\;\;\bigtriangledown \times \vec{f},\;\;\;\bigtriangledown^{-1} \cdot \vec{f}\;\;\;and\;\;\;\bigtriangledown^{-1} \times \vec{f}[/tex]

    What you think about all this?
  2. jcsd
  3. Dec 11, 2013 #2

    Simon Bridge

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    Science Advisor
    Homework Helper

    It may make more sense of you expand \vec r in a basis - say: Cartesian.
    See which ideas make sense.

    But you can differentiate and integrate vectors - look up vector-valued functions for instance.
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