1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Please just check this [Analysis problem]

  1. Nov 10, 2004 #1

    AKG

    User Avatar
    Science Advisor
    Homework Helper

    Find the (first-order) partial derivatives of the following function (where g : [itex]\mathbb{R} \to \mathbb{R}[/itex] is continuous):

    [tex]f(x, y) = \int _a ^{\int _b ^y g} g.[/tex]

    --------------

    I got:

    [tex]D_1f(x, y) = 0[/tex]

    [tex]D_2f(x, y) = \frac{\partial f}{\partial y}[/tex]

    [tex]= \frac{\partial }{\partial y} \int _a ^{\int _b ^y g}g[/tex]

    Let G be the antiderivative of g:

    [tex]= \frac{\partial }{\partial y}\left ( G \left (\int _b ^y g\right ) - G(a) \right )[/tex]

    [tex]= \frac{\partial }{\partial y} G \left (\int _b ^y g\right )[/tex]

    [tex]= \left ( \frac{\partial G \left (\int _b ^y g\right )}{\partial \left (\int _b ^y g\right ) }\right ) \left (\frac{\partial }{\partial y}\int _b ^y g\right )[/tex]

    [tex]= g\left (\int _b ^y g\right )\frac{\partial }{\partial y}\left (G(y) - G(b) \right )[/tex]

    [tex] = g\left (\int _b ^y g\right )g(y)[/tex]
     
    Last edited: Nov 10, 2004
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted



Similar Discussions: Please just check this [Analysis problem]
Loading...