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Partial differentiation and changing variables

  1. Nov 29, 2003 #1
    Maths Question: I am having a lot of problems with this question, can any undergrad physicists or mathematicians help me?

    (note: p before a differntial= partial derivative) .

    Spherical polar coordinates (r, (thetha), (phi)) are defined in terms of Cartesian coorindates (x,y,z) by:

    x=rsin(theta)cos(phi)
    y=rsin(theta)sin(phi)
    z=rcos(theta)

    given that f is a function of r only, independent of theta and phi, show that

    p(df)/p(dx) = (x/r).(df/dr)

    p(d^2f)/p(dx^2) = (1/r).(df/dr) + (x^2/r).d[(1/r).(df/dr)]/dr

    and hence deduce that:

    p(d^2f)/p(dx^2) + p(d^2f)/p(dy^2) + p(d^2f)/p(dz^2) =
    (1/r^2).d[r^2.(df/dr)]/dr


    a) is straigthforward, any thoughts on how to appraoch b) ???

    thanks
     
  2. jcsd
  3. Nov 29, 2003 #2

    jamesrc

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    Consider the fact that x2 + y2 + z2 = r2

    What you've already shown for x also applies to y and z:

    [itex] \frac{\partial f}{\partial x} = \frac{x}{r} \cdot \frac{df}{dr} [/itex]

    [itex] \frac{\partial f}{\partial y} = \frac{y}{r} \cdot \frac{df}{dr} [/itex]

    [itex] \frac{\partial f}{\partial z} = \frac{z}{r} \cdot \frac{df}{dr} [/itex]

    [itex] \frac{\partial^2 f}{\partial x^2} = \frac{1}{r} \cdot \frac{df}{dr} + \frac{x^2}{r}\,\frac{d}{dr}\left(\frac{1}{r}\,\frac{df}{dr}\right) [/itex]

    [itex] \frac{\partial^2 f}{\partial y^2} = \frac{1}{r} \cdot \frac{df}{dr} + \frac{y^2}{r}\,\frac{d}{dr}\left(\frac{1}{r}\,\frac{df}{dr}\right) [/itex]

    [itex] \frac{\partial^2 f}{\partial z^2} = \frac{1}{r} \cdot \frac{df}{dr} + \frac{z^2}{r}\,\frac{d}{dr}\left(\frac{1}{r}\,\frac{df}{dr}\right) [/itex]

    Add them together and use the first equation to get:

    [itex] \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} = r\frac{d}{dr}\left(\frac{1}{r}\,\frac{df}{dr}\right) + \frac{3}{r}\frac{df}{dr} [/itex]

    which can be simplified to:

    [itex] \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} = \frac{d^2 f}{dr^2} + \frac{2}{r}\frac{df}{dr} = \frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{df}{dr}\right) [/itex]
     
  4. Nov 29, 2003 #3
    thanks

    Thanks so much, you've made it very clear!
     
  5. Dec 31, 2003 #4
    Hi,

    I would much appreciate some help in deriving the second expression in the original question below:

    p(d^2f)/p(dx^2) = (1/r).(df/dr) + (x^2/r).d[(1/r).(df/dr)]/dr

    Thanks for the help!

    Regards,

    Sam
     
  6. Dec 31, 2003 #5

    jamesrc

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    Use the product rule to find the derivative of this equation:

    [itex] \frac{\partial f}{\partial x} = \frac{x}{r} \cdot \frac{df}{dr} [/itex]
     
  7. Dec 31, 2003 #6
    Thanks for the reply,

    I am trying this but am getting very confsed...


    first of all:

    I have a term: dr/dx .dx/dr. But I dont see why this is equal to one. If i accept this I get the result, but we know:

    x = r.cos(phi).sin(theta)

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

    so we can verify that the 2 partial derivatives are not eqal.

    secondly, f is a function of r ony.... but df/dr (partial) is also a function of p and theta...

    Therefore, you have a lot more terms than you would expect, because you have to take the derivative wrt phi and theta and then these wrt to x.

    I dont know if I have completely misunderstood partial derivatives, but this is gettig messy!

    Regards,

    Sam
     
  8. Dec 31, 2003 #7

    jamesrc

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    You are given that f = f(r) (function of r only).

    Using the chain rule:
    [tex] \frac{\partial f}{\partial x} = \frac{df}{dr}\cdot\frac{\partial r}{\partial x} [/tex]

    (notice that it's df/dr (not partials) because f is a function of r only)

    Now we need to find [tex] \frac{\partial r}{\partial x} [/tex].

    To do this, you can use the fact that x2 + y2 + z2 = r2.

    Take the derivative of both sides with respect to x to get:

    [tex] 2x = 2r\frac{\partial r}{\partial x} [/tex]

    rearranging:

    [tex] \frac{\partial r}{\partial x} = \frac{x}{r} [/tex]

    plug that in to the results from the chain rule:

    [tex] \frac{\partial f}{\partial x} = \frac{x}{r}\cdot\frac{df}{dr} [/tex]

    Okay so far?

    Now on to the second part using the product rule:

    [tex] \frac{\partial^2 f}{\partial x^2} = x\frac{\partial}{\partial x}\left(\frac{1}{r}\cdot\frac{df}{dr} \right) + \frac{1}{r}\cdot\frac{df}{dr} [/tex]

    from before: [tex] \partial x = \frac{r}{x}\partial r [/tex]

    substitute that in to get:

    [tex] \frac{\partial^2 f}{\partial x^2} = x\frac{x}{r}\frac{\partial}{\partial r}\left(\frac{1}{r}\cdot\frac{df}{dr} \right) + \frac{1}{r}\cdot\frac{df}{dr} [/tex]

    which becomes:

    [itex] \frac{\partial^2 f}{\partial x^2} = \frac{1}{r} \cdot \frac{df}{dr} + \frac{x^2}{r}\,\frac{d}{dr}\left(\frac{1}{r}\,\frac{df}{dr}\right) [/itex]

    Hope that helps.
     
  9. Jan 1, 2004 #8
    Much Appreciated.

    Sam
     
  10. Oct 3, 2004 #9

    wai

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    the function f(x,y) is transformed to F(r,a) by change of variables x=rcos a and y=rsin a. Show that
    1. p(d^2F)/p(dr^2)
    2. p(d^2F)/p(da^2)+r[p(dF/dr)]
     
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