Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Chain rule, transforming PDE

  1. Nov 14, 2009 #1
    If [tex]z = f(x,y)[/tex] and [tex]x = r \cos{v}[/tex], [tex] y = r\sin{v}[/tex] the object is to show that [tex] d = \partial [/tex] since it's easier to do on computer

    Show that:
    [tex] \frac{d^2 z}{dr^2} + \frac{1}{r} \frac{dz}{dr} + \frac{1}{r^2} \frac{d^2 z}{dv^2} = \frac{d^2 z}{dx^2} + \frac{d^2 z}{dy^2} [/tex]

    It's from Adams calculus, will show where I get lost.

    We have from the chain rule
    [tex] \frac{dz}{dr} = \frac{dz}{dx} \frac{dx}{dr} + \frac{dz}{dy} \frac{dy}{dr} [/tex]

    But [tex] \frac{dx}{dr} = \cos{v} [/tex] and [tex] \frac{dy}{dr} = \sin{v} [/tex], so gives:

    [tex] \frac{dz}{dr} = \frac{dz}{dx} \cos{v} + \frac{dz}{dy} \sin{v} [/tex]

    Now we need [tex] \frac{d^2 z}{dr^2} [/tex], where I end up in trouble. We have

    [tex] \frac{d^2 z}{dr^2} = \frac{d}{dr} \cos{v} \frac{dz}{dx} + \frac{d}{dr} \sin{v} \frac{dz}{dy} [/tex] which can be written as

    [tex] \frac{d^2 z}{dr^2} = \cos{v} \frac{d}{dr} \frac{dz}{dx} + \sin{v} \frac{d}{dr} \frac{dz}{dy} [/tex]

    Now my books says that

    [tex] \frac{d}{dr} \frac{dz}{dx} = \cos{v} \frac{d^2 z}{dx^2} + \sin{v} \frac{d^2 z}{dy dx} [/tex] if I understand correct. But I don't see how you get to this ...
     
  2. jcsd
  3. Nov 14, 2009 #2
    Since z is a function of (x,y), the derivative of z in respect to x (or y), is another function of (x,y). Therefore the operator d/dr acts on the derivative of z the same way it acts on z.

    Look at this:
    [tex]\frac{d}{dr}(z)=\frac{dx}{dr}\frac{d}{dx}(z)+\frac{dy}{dr}\frac{d}{dy}(z)=[/tex]

    Substitude z by dz/dx and you get the final result which you have written.
     
  4. Nov 14, 2009 #3
    Thank you very much, the equation for d^2 z/dv^2 got even messier. But I think I understand, I solved it and I'm going to try other examples on this.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Chain rule, transforming PDE
  1. Chain Rule (Replies: 9)

  2. Chain Rule (Replies: 30)

  3. Chain Rule (Replies: 25)

  4. Chain rule. (Replies: 6)

  5. The Chain Rule (Replies: 3)

Loading...