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I Fixed Variables in Partial Derivatives

  1. Jan 24, 2017 #1
    My physics book is showing an example of why it matters "what variable you hold fixed" when taking the partial derivative. So it asks to show that
    ##(\frac{\partial{w}}{\partial{x}})_{y} \neq (\frac{\partial{w}}{\partial{x}})_z##
    where ##w=xy## and ##x=yz## and the subscripts are what variable is held fixed.

    I'm just not sure what it means by holding a variable "fixed" since all other variables except the two in question are treated as constants, so why would it matter whether ##w## is a function of ##(x, y)## or a function of ##(x, z)##
    I mean I see that the partial derivatives don't match up, but don't really see why the variables make a difference since they are the same function.
     
  2. jcsd
  3. Jan 25, 2017 #2

    Orodruin

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    The condition ##x=yz## describes a surface in three dimensions. Now, you can use different sets of variables to parametrise this surface. One choice is to use x and y and another to use x and z. Varying x in the first parametrisation is generally going to give you a displacement in the surface that will be different from the one you obtain if you vary x in the same way in the second parametrisation. The "keeping these variables fixed" essentially tells you what parametrisation is used and therefore exactly what variations should be considered.
     
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