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Partial derivatives - verify solution?

  1. Mar 5, 2008 #1
    [SOLVED] partial derivatives - verify solution?

    Let [tex]f:\mathbb{R}^3\rightarrow\mathbb{R}[/tex], [tex]g:\mathbb{R}^2\rightarrow\mathbb{R}[/tex], and [tex]F:\mathbb{R}^2\rightarrow\mathbb{R}[/tex] be given by
    [tex]F(x,y)=f(x,y,g(x,y))[/tex].
    1. Find DF in terms of the partial derivatives of f and g.
    2. If F(x,y)=0 for all (x,y), find [tex]D_1g[/tex] and [tex]D_2g[/tex] in terms of the partial derivatives of f.

    My solution:
    1. [tex]DF=D_1F+D_2F=(f_1+f_3g_1)+(f_2+f_3g_2)[/tex]
    2. If [tex]f_3\neq0[/tex], then we have the partials of F being zero, so:
    [tex]g_1=-f_1/f_3[/tex] and [tex]g_2=-f_2/f_3[/tex]. However, if [tex]f_3=0[/tex] then we have [tex]f_1=f_2=0[/tex].

    My concern is with the last part of 2. If [tex]f_3=0[/tex], then I cannot make any statement about the partials of g. Am I doing something wrong?


    NOTE: [tex]f_1[/tex] refers to differentiation of f by the first variable.
     
  2. jcsd
  3. Mar 5, 2008 #2

    HallsofIvy

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    No, that's completely true. If F is, in fact, NOT a function of g, then no information about F can tell you anything about g!
     
  4. Mar 5, 2008 #3
    Thanks! :)
     
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