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Laplace in Cyl. form: one step I'm not sure about

  1. May 19, 2015 #1
    I have completed the exercise, but I did something weird in one step to make it work, and I'd like to know more about what I did...or if what I did was at all valid.

    1. The problem statement, all variables and given/known data


    Show that Laplace's equation

    [tex]\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}=0[/tex]

    when converted to cylindrical coordinates gives

    [tex]\frac{\partial^2u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{\partial^2u}{\partial \theta^2}+\frac{\partial^2u}{\partial z^2}=0[/tex]

    2. Relevant equations

    obviously

    [tex]x = r\cos{\theta}[/tex]
    [tex]y = r\sin{\theta}[/tex]
    [tex]z = z[/tex]

    3. The attempt at a solution

    It was a long exercise, so I'll skip ahead to just before the step I'm asking about

    [tex]r^2\frac{\partial^2u}{\partial r^2}+\frac{\partial^2u}{\partial \theta^2}+\frac{\partial^2u}{\partial z^2} =
    \frac{\partial^2u}{\partial x^2}(r^2\cos^2{\theta}+r^2\sin^2{\theta})+\frac{\partial^2u}{\partial y^2}(r^2\cos^2{\theta}+r^2\sin^2{\theta})+\frac{\partial u}{\partial x}(-r\cos{\theta})+\frac{\partial u}{\partial y}(-r\sin{\theta})+\frac{\partial^2u}{\partial z^2}[/tex]

    [tex]r^2\frac{\partial^2u}{\partial r^2}+\frac{\partial^2u}{\partial \theta^2} =
    r^2\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}\right)+\frac{\partial u}{\partial x}(-r\cos{\theta})+\frac{\partial u}{\partial y}(-r\sin{\theta})[/tex]

    Now, I got rid of the second partial of u to z since it just takes up space, and is not interesting right now. What I am interested in is the last two terms of the RHS. By the chain rule,

    [tex]\frac{\partial u}{\partial x}(-r\cos{\theta})+\frac{\partial u}{\partial y}(-r\sin{\theta}) = \frac{\partial u}{\partial r}\frac{\partial r}{\partial x}(-r\cos{\theta})+\frac{\partial u}{\partial r}\frac{\partial r}{\partial y}(-r\sin{\theta})[/tex]

    [tex]\frac{\partial r}{\partial x} = \frac{1}{\cos{\theta}} \ and \ \frac{\partial r}{\partial y} = \frac{1}{\sin{\theta}}[/tex]

    So

    [tex]\frac{\partial u}{\partial x}(-r\cos{\theta})+\frac{\partial u}{\partial y}(-r\sin{\theta}) = -r\left(\frac{\partial u}{\partial r}+\frac{\partial u}{\partial r}\right)[/tex]

    Now...here's where I need more insight, I have an inkling that the following is NOT right

    [tex]\frac{\partial u}{\partial x}(-r\cos{\theta})+\frac{\partial u}{\partial y}(-r\sin{\theta}) = -2r\frac{\partial u}{\partial r}[/tex]

    My intuition that this was wrong came from a few places. First off, it did not look like it would give the solution I was being directed to find...but that isn't sufficient for real understanding.

    My understanding is that when using partial derivatives, especially when using them with the chain rule, one is finding rates of change in given directions, thus it's kind of like a vector (or basis vector - I haven't studied linear algebra yet) situation...and one cannot simply add basis vectors to each other...so I opted for this

    [tex]\frac{\partial u}{\partial x}(-r\cos{\theta})+\frac{\partial u}{\partial y}(-r\sin{\theta}) = -r\frac{\partial u}{\partial r}[/tex]

    and of course, by reinserting the second partial for u on z and rearranging one can show the very thing the problem asked us to show.

    However, it bothers me that I have

    [tex]-r\left(\frac{\partial u}{\partial r}+\frac{\partial u}{\partial r}\right) = -r\frac{\partial u}{\partial r}[/tex]

    What is going on here?

    Like I said, I've completed the exercise, but I feel like my insight is lacking.
     
  2. jcsd
  3. May 19, 2015 #2

    Orodruin

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    Your partial derivatives ##\partial r / \partial x## and ##\partial r/\partial y## are wrong. You cannot simply solve for ##r## from ##x = r\cos\theta## and differentiate wrt x, ##\theta## depends on x and y! You need to differentiate the full expression which defines r in terms of x and y, i.e., ##r^2 =x^2+y^2##.
     
  4. May 21, 2015 #3
    So even though we can consider theta as being constant when we partial differentiate [itex]\frac{\partial x}{\partial r}[/itex] or [itex]\frac{\partial y}{\partial r}[/itex], I need to understand that [itex]\frac{\partial r}{\partial x}[/itex] is not simply the reciprocal of [itex]\frac{\partial x}{\partial r}[/itex]. So is this what makes PDE's so difficult? Because this then suggests to me that only ordinary differential fractions are separable...I haven't done anything beyond ODE's yet. Is the ability to flip and separate the differential fraction what they mean by homogenous?
     
  5. May 21, 2015 #4

    Orodruin

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    If by this you mean that ##dy/dx = 1## implies ##dy = dx##, then yes, this is not generally true when you have several variables. If you have several ##y## and several ##x##, you will have
    $$
    dy^i = \frac{\partial y^i}{\partial x^j} dx^j
    $$
    (here using Einstein summation convention, the ##j## is a summation index). When we talk about a separable PDE, we talk about a PDE which can be broken down into a set of ODEs by using separation of variables, e.g., ##u(x,t) = X(x)T(t)##.
     
  6. May 21, 2015 #5
    Basically the chain rule with partial derivatives is abit different, it is ##\frac{\partial u}{\partial r}=\frac{\partial u}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial r}## and similar for theta. Using this expression you can easily see that ##-r\frac{\partial u}{\partial r}## gives the desired result.

    This is from a theorem in calculus regarding the partial derivation of a composition of function of 2 variables with 2 other functions of 2 variables each (just as the chain rule with ordinary derivatives is from the composition of a function of 1 variable with another function of 1 variable).

    Here our function of two variables is ##u(x,y)## and we take the composition of this function with the functions ##x=x(r,\theta)=rcos\theta## and ##y=y(r,\theta)=rsin\theta##

    So basically your mistake is when you say that ##\frac{\partial u}{\partial x}=\frac{\partial u}{\partial r}\frac{\partial r}{\partial x}##, this is not correct, the correct is ##\frac{\partial u}{\partial x}=\frac{\partial u}{\partial r}\frac{\partial r}{\partial x}+\frac{\partial u}{\partial \theta}\frac{\partial \theta}{\partial x}## with ##r^2=x^2+y^2## and ##\theta=arctan(y/x)##.
     
    Last edited: May 21, 2015
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