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Math Help - help with total differentiation

  1. #1
    Junior Member
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    help with total differentiation

    Hello,

    If w=f(ax+by) I am asked to show -please read w[x] as partial derivative fo w w.r.t.x- that: bw[x]-aw[y]=0 this tells me that the difference of a ccahnge in the coordinates of a points on on a curve in different directiosn will be zero (though pls dont be too harsh if I have - which i probably have - got the whole geometry thing wrong

    i am given as a hint to let ax +by = z

    i can totally differentiate this : dz = adx + bdy

    and totally differentiated w: dw = w[x]dx + w[y]dy but don't know hyow to come up with the desried result.

    pls help
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  2. #2
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    Quote Originally Posted by pepsi View Post
    Hello,

    If w=f(ax+by) I am asked to show -please read w[x] as partial derivative fo w w.r.t.x- that: bw[x]-aw[y]=0 this tells me that the difference of a ccahnge in the coordinates of a points on on a curve in different directiosn will be zero (though pls dont be too harsh if I have - which i probably have - got the whole geometry thing wrong

    i am given as a hint to let ax +by = z

    i can totally differentiate this : dz = adx + bdy

    and totally differentiated w: dw = w[x]dx + w[y]dy but don't know hyow to come up with the desried result.

    pls help

    You have that w(z)=f(z)\,,\; with\; z=ax+by, so applying the chain rule:

    w_x=\frac{\partial f}{\partial z}\,\frac{\partial z}{\partial x}=a\cdot \frac{\partial f}{\partial z}

    w_y=\frac{\partial f}{\partial z}\,\frac{\partial z}{\partial y}=b\cdot \frac{\partial f}{\partial z}

    Well, now multiply the first one above by b and the second one by a, substract and get zero...

    Tonio
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