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Math Help - Functions of two variables and small increments.

  1. #1
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    Functions of two variables and small increments.

    Okay,

    I understand that if a differentiable function z=f(g(x,y)) and x, y are the independent variables, then

    \frac{\partial{g}}{\partial{x}} = \lim_{\delta{x}\to{0}}\left(\frac{\delta{g}}{\delt  a{x}}\right) and \frac{\partial{f}}{\partial{g}} = \lim_{\delta{g}\to{0}}\left(\frac{\delta{f}}{\delt  a{g}}\right), so \frac{\partial{f}}{\partial{x}} = \frac{\partial{f}}{\partial{g}}.\frac{\partial{g}}  {\partial{x}}, keeping y constant

    and

    \frac{\partial{g}}{\partial{y}} = \lim_{\delta{y}\to{0}}\left(\frac{\delta{g}}{\delt  a{y}}\right) and \frac{\partial{f}}{\partial{g}} = \lim_{\delta{g}\to{0}}\left(\frac{\delta{f}}{\delt  a{g}}\right), so \frac{\partial{f}}{\partial{y}} = \frac{\partial{f}}{\partial{g}}.\frac{\partial{g}}  {\partial{y}}, keeping x constant

    now, if I let u=g(x, y) can someone show/explain using the above method what \frac{\partial{z}}{\partial{u}} would be?.

    Thanks.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by bluntpencil View Post
    Okay,

    I understand that if a differentiable function z=f(g(x,y)) and x, y are the independent variables, then

    \frac{\partial{g}}{\partial{x}} = \lim_{\delta{x}\to{0}}\left(\frac{\delta{g}}{\delt  a{x}}\right) and \frac{\partial{f}}{\partial{g}} = \lim_{\delta{g}\to{0}}\left(\frac{\delta{f}}{\delt  a{g}}\right), so \frac{\partial{f}}{\partial{x}} = \frac{\partial{f}}{\partial{g}}.\frac{\partial{g}}  {\partial{x}}, keeping y constant

    and

    \frac{\partial{g}}{\partial{y}} = \lim_{\delta{y}\to{0}}\left(\frac{\delta{g}}{\delt  a{y}}\right) and \frac{\partial{f}}{\partial{g}} = \lim_{\delta{g}\to{0}}\left(\frac{\delta{f}}{\delt  a{g}}\right), so \frac{\partial{f}}{\partial{y}} = \frac{\partial{f}}{\partial{g}}.\frac{\partial{g}}  {\partial{y}}, keeping x constant

    now, if I let u=g(x, y) can someone show/explain using the above method what \frac{\partial{z}}{\partial{u}} would be?.

    Thanks.
    If, as you say, z= f(g(x,y)) and u= g(x,y), then you have z= f(u) and there are no "partial" derivatives. You should be writing \frac{dz}{du}.

    \frac{dz}{du}= \frac{dz}{du} since "u" and "g" are equal- they are just different ways of writing the same thing.

    In fact, you should not be writing " \frac{\partial f}{\partial g}" because f is a function of the single variable g.

    The correct chain rules are \frac{\partial f}{\partial x}= \frac{df}{dg}\frac{\partial g}{\partial x} and \frac{\partial f}{\partial y}= \frac{df}{dg}\frac{\partial g}{\partial y}.
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  3. #3
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    I stand corrected. I never could get the hang of the notation, until now. This has helped me a lot in rates of change and change of variable problems.

    Much appreciated.
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