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What exactly means (∂^2/∂x ∂y) ?

  1. Nov 8, 2011 #1
    what exactly means (∂^2/∂x ∂y) ??

    Not sure if I should post it here or in the homework section.
    feel free to move the topic if necessary.


    Due to some personal reasons I'm not attending college this semester. So I'm studying using the just books, which is kinda good but I have no one to ask questions (in person that is).

    So I'm sorry if I make some trivial questions and all.


    Anyways:

    This is something that has been bothering me.
    what exactly is [itex] \frac{\partial^2 f }{\partial x \partial y}[/itex] and what is it application.

    what I don`t understand is:
    when taking the first partial derivative you treat y as a constant and x as a variable.
    then why is it that you do the opposite when taking the 2nd derivative?

    What exactly does it represent?
     
  2. jcsd
  3. Nov 8, 2011 #2

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    Re: what exactly means (∂^2/∂x ∂y) ??

    First off, like many definitions in mathematics, the definition exists because it happens to be useful. If it wasn't useful we wouldn't use it.

    So what is it useful for? Think of a Taylor series expansion of some function of a single variable,

    [tex]f(x) = f(x_0)
    + (x-x_0)\,\left.\frac{df(x)}{dx}\right|_{x=x_0}
    + \frac 1 {2!} (x-x_0)^2 \, \left. \frac{d^2f(x)}{dx^2} \right|_{x=x_0}
    + \cdots[/tex]

    How to generalize this to a function of multiple variables? The answer lies in these partial derivatives:

    [tex]f(\vec x) = f(\vec x_0)
    + (\vec x-\vec x_0)^T \left.\nabla f(\vec x) \right|_{x=x_0}
    + \frac 1 {2!} (\vec x-\vec x_0)^T \left.\boldsymbol H(\vec x)\right|_{x=x_0}
    (\vec x-\vec x_0)
    + \cdots[/tex]

    where [itex]\nabla f(\vec x)[/itex] is the gradient of f(x) with respect to x and [itex]\boldsymbol H(\vec x)[/itex] is the Hessian of f(x) with respect to x. The elements of the gradient are the first partial derivatives while the elements of the Hessian are the second partials.
     
  4. Nov 8, 2011 #3

    Mark44

    Staff: Mentor

    Re: what exactly means (∂^2/∂x ∂y) ??

    In words, this symbol is the partial derivative with respect to x of the partial derivative with respect to y.

    Here f would be a function of x and y.

    Example: f(x, y) = x2 + 3xy
    [itex]\frac{\partial f}{\partial y} = 3x[/itex]
    [itex]\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = 3[/itex]
    When you're working with partial derivatives, the phrase "first partial derivative" doesn't make much sense, because you haven't said which variable you are differentiating with respect to.

    For a function of two variables, there are two first partials: the partial with respect to x, [itex]\frac{\partial f}{\partial x}[/itex] or fx, and the partial with respect to y, [itex]\frac{\partial f}{\partial y}[/itex] or fy. There are four second partials, which are obtained by differentiating each of the two first partials with respect to either x or y.
    A first partial represents the rate of change of the function value relative to change in the x or y direction.

    A second partial represents the rate of change of the value of one of the first partials relative to change in the x or y direction.
     
  5. Nov 8, 2011 #4
    Re: what exactly means (∂^2/∂x ∂y) ??

    You need to draw some pictures Renko. Tell you what, I bet you could create a demonstration for the Wolfram demonstration project for this. Take the function:

    [tex]f(x,y)=3xy^2+y^3x+4x^3y+2y[/tex]

    Plot it nice in Mathematica. Now compute the partial with respect to x but do more than that. On the plot, draw a line at the point (x,y) which has the slope of:

    [tex]\frac{\partial f}{\partial x}[/tex]

    Encapsulate this in a Manipulate function so as you slide the slider across the function for a constant y, this line, say yellow, move across the function surface showing what the partial with respct to x is at a particular point x. Now go a step further and compute the rate at which the slope of thay yellow line changes as you vary y. That's the mixed-partial:

    [tex]\frac{\partial}{\partial y} \left(\frac{\partial f}{\partial x}\right)[/tex]

    Yeah, make a slider for that one too:

    So as you change the y-slider, that is as you move the value of y, the rate at which that yellow line is moving up or down with change according to the value of the mixed partial.

    Be a nice project for you and sides, you get your name published for all the world to see.
     
    Last edited: Nov 8, 2011
  6. Nov 10, 2011 #5
    Re: what exactly means (∂^2/∂x ∂y) ??


    Of Course!
    Now I understand why I was confused.

    I was thinking in 3 dimensions when dealing with the functon and its partial derivatives.
    But for some reason I was thinking in 2 dimensions when dealing with its second partial derivatives.

    thinking now, it seems kinda obvious that a function of 2 variables HAS to have four partial derivatives.
     
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