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Math Help - Vector Differentiation

  1. #1
    Member kjchauhan's Avatar
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    Vector Differentiation

    Please help me to solve this problem:

    Prove that:

    \nabla^2 f(r) = f''(r) + \frac{2}{r}f'(r)

    Where r^2=x^2+y^2+z^2

    Thanks in advance..
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  2. #2
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    Hint:

    \displaystyle \frac{\partial f(r)}{\partial x} = f'(r) \frac{\partial r}{\partial x} = f'(r)\frac{x}{r}

    \displaystyle \frac{\partial^2 f(r)}{\partial^2 x} = \frac{\partial^2 f'(r)\frac{x}{r}}{\partial x} = f''(r)\frac{x^2}{r^2} + f'\frac{r^2 - x^2}{r^3}.
    Last edited by snowtea; January 10th 2011 at 07:51 AM.
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  3. #3
    Member kjchauhan's Avatar
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    Quote Originally Posted by snowtea View Post
    Hint:

    \displaystyle \frac{\partial f(r)}{\partial x} = f'(r) \frac{\partial r}{\partial x} = f'(r)\frac{x}{r}

    \displaystyle \frac{\partial^2 f(r)}{\partial^2 x} = \frac{\partial^2 f'(r)\frac{x}{r}}{\partial x} = f''(r)\frac{x^2}{r} + f'\frac{r^2 - x^2}{r^3}.
    Thanks for Hint..
    This is correct but I couldn't find the right hand side ..
    According to your hint, the first term of the right will be rf''(r)..
    So, is it correct??

    Ok, well, got it..

    Thank you very much..
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  4. #4
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    Quote Originally Posted by kjchauhan View Post
    Thanks for Hint..
    This is correct but I couldn't find the right hand side ..
    According to your hint, the first term of the right will be rf''(r)..
    So, is it correct??
    I made a typo. It should be \frac{x^2}{r^2}. It is now fixed in the previous post.
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