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Math Help - Vector Differentiation etc.

  1. #1
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    Vector Differentiation etc.

    Given that V = xyz(r^−7), find the first partial derivative of V (wrt x) and deduce that the second partial derivative of V (wrt to x) is

    −21xyz(r^−9) + (63x^3)yz(r^−11)

    NB: d = delta sign, del = del function (upside down triangle)

    I've tried it several times now. I can see where the coefficents will eventually come from being multiples of 7, but I can't get the answer above. Can anyone help? Thanks.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Yerobi View Post
    Given that V = xyz(r^−7), find the first partial derivative of V (wrt x) and deduce that the second partial derivative of V (wrt to x) is

    −21xyz(r^−9) + (63x^3)yz(r^−11)

    NB: d = delta sign, del = del function (upside down triangle)

    I've tried it several times now. I can see where the coefficents will eventually come from being multiples of 7, but I can't get the answer above. Can anyone help? Thanks.
    What is r? Is it r = \sqrt{x^2 + y^2 + z^2}?

    -Dan
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  3. #3
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    Indeed.
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  4. #4
    Senior Member Peritus's Avatar
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    a preliminary calculation:

    r = \sqrt {x^2  + y^2  + z^2 }  \Rightarrow \frac{{\partial r}}<br />
{{\partial x}} = \frac{x}<br />
{r}

    -------------------------------------------------

    \begin{gathered}<br />
  V = xyzr^{ - 7}  \hfill \\<br />
   \hfill \\<br />
  \frac{{\partial V}}<br />
{{\partial x}} = yzr^{ - 7}  - 7xyzr^{ - 8} \frac{{\partial r}}<br />
{{\partial x}} = yzr^{ - 7}  - 7x^2 yzr^{ - 9}  \hfill \\ <br />
\end{gathered}

    thus:

    <br />
\frac{{\partial ^2 V}}<br />
{{\partial x^2 }} =  - 7yzr^{ - 8} \frac{{\partial r}}<br />
{{\partial x}} - 14xyzr^{ - 9}  + 63x^2 yzr^{ - 10} \frac{{\partial r}}<br />
{{\partial x}} =  - 21xyzr^{ - 9}  + 63x^3 yzr^{ - 11} <br />
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  5. #5
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    Thanks! The next part of the Q is:

    Use symmetry to write down the corresponding expressions for the second derivatives of V wrt to y and z.

    Deduce that (del)^2(V) = 0.

    -----

    The second derivative of V wrt y would be the same but using dr/dx = y/r etc. yeah? Once I have derivatives wrt y and z, would their sum equal 0?
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