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I Symbol used in total differential and small errors

  1. Mar 27, 2016 #1
    Mod note: Thread title changed from "Sign used in total differential and small errors" to "Symbol used in total differential and small errors"
    My notes say that z=f(x,y) can be defined as the sum of the changes with respect to x,y respectively. The formula is:

    dz = ∂f/∂x * dz + ∂f/∂y * dy

    Whereas in small errors, the formula is:

    deltaf = ∂f/∂x * deltax + ∂f/∂y * deltay

    Shouldn't the sign used be ∂f since it is taken with respect to more than one variable?
     
    Last edited by a moderator: Mar 27, 2016
  2. jcsd
  3. Mar 27, 2016 #2

    Mark44

    Staff: Mentor

    Both the text of your notes and the formula above are incorrect. "z = f(x, y)" is not defined as a sum. The differential of z, dz, is defined as a sum. The formula you wrote has a typo, and should be dz = ∂f/∂x * dx + ∂f/∂y * dy or df = ∂f/∂x * dx + ∂f/∂y * dy.
    No. The actual change in f is ##\Delta f##, which is only approximately equal to ##\frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y##.

    ∂f by itself has no meaning here, but df does. In the calculation of error, you use df to approximate the actual change in function value, ##\Delta f##.

    If you know ##\Delta x## and ##\Delta y##, then ##\Delta z \approx dz = \frac{\partial z}{\partial x} \Delta x + \frac{\partial z}{\partial y} \Delta y##.
     
  4. Apr 1, 2016 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Surely you meant to say that dz, not z, can be defined that way.
     
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