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I Partial Vector Derivative

  1. Oct 15, 2017 #1
    Is this the correct partial derivative of B?

    ##\vec{B} = \frac{g \vec{r}}{4 \pi r^3}##

    ##\frac{\partial \vec B}{\partial r}## = ##-3\frac{g \vec{r}}{4 \pi r^4} + \frac{g}{4 \pi r^3 }(\frac{\partial r_r \hat r}{\partial r})##
     
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  3. Oct 15, 2017 #2

    Orodruin

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    What is ##r_r##?

    Apart from that it seems as if you are just applying the product rule for derivatives.
     
  4. Oct 15, 2017 #3
    ##r_r## is the radial part of a vector from the origin to an arbitrary point to be examined:

    ##\vec{r} = r_r \hat r +r_\theta \hat \theta +r_\phi \hat \phi##
     
  5. Oct 15, 2017 #4

    Orodruin

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    Then yes and no. Yes because it is technically correct due to ##r_r = r## and ##r_\theta = r_\phi = 0##. No since you generally cannot assume that ##\partial_r \vec w = \partial_r w_r \hat r##, the general expression is ##\partial_r \vec w = \partial_r (w_r\hat r + w_\theta \hat \theta + w_\phi \hat \phi)## and doing so generally will get you the incorrect result.
     
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