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Question about partial derivatives.

  1. Oct 6, 2015 #1
    I have a multivariable function z = x2 + 2y2 such that x = rcos(t) and y = rsin(t). I was asked to find upload_2015-10-6_20-22-15.png (I know the d's should technically be curly, but I am not the best at LaTeX). I thought this would just be a simple application of chain rule:
    2/(∂y∂t) = (∂z/∂x)(ⅆx/ⅆt) + (∂z/∂y)(ⅆy/ⅆt)
    but apparently this is not the case. Could someone perhaps explain why this is not the right thing to do. When I did it this way I got 2x = 2rcos(t) = 2ycot(t) as my answer, but my book says the answer is -2y2cot(t)csc2(t) and I have no clue what they are doing.

    Any help would be appreciated. Thank you in advance.
     
  2. jcsd
  3. Oct 6, 2015 #2

    SteamKing

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    Staff Emeritus
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    The derivative, ## \frac {∂^2z}{∂y∂t} ## is known as a mixed partial derivative. Since the expression contains only the function z differentiated w.r.t. y and t, there is no need to differentiate z w.r.t. x. You may differentiate z w.r.t. y first, then w.r.t. t next.
     
  4. Oct 6, 2015 #3
    @SteamKing But does it not matter that x is a function of t, and (through the relation with r) a function of y? And the only way the book's answer makes sense is if you substituted rcos(t) = ycot(t) into x^2 then differentiate that, treating 2y^2 as a constant, which knocks it out completely.
     
  5. Oct 6, 2015 #4

    Mark44

    Staff: Mentor

    I'm confused as to what you're doing. In the first paragraph, you write ##\frac{\partial^2 z}{\partial t \partial y}##, but later you write the opposite order, ##\frac{\partial^2 z}{\partial y \partial t}##. The first mixed partial is the same as this: ##\frac{\partial }{\partial t} \frac{\partial z}{\partial y}##. In the second mixed partial, the differentiation occurs in the opposite order.

    Here's the LaTeX I used, tweaked a bit so that it won't render:
    ##\frac{\partial^2 z}{\partial t \partial y}##
    ##\frac{\partial^2 z}{\partial y \partial t}##
    ##\frac{\partial }{\partial t} \frac{\partial z}{\partial y}##
     
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