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I Differentiation is Exact or Approximation

  1. Aug 5, 2016 #1
    Is Differentiation exact or just an approximation?

    I am wonder whether this question is meaningful or not. Slope is expressed as "it is approaching to a value as x is approaching 0" so it is inappropriate to ask such question. But when I deal with uniform circular motion, it is very confusing.

    Suppose ##A## is constant for vector ##\mathbf A##. And the angle between ##\mathbf A(t+\Delta t)## and ##\mathbf A(t)## is ##\Delta \theta##.
    ##\begin{align}
    \Delta \mathbf A & = \mathbf A (t + \Delta t) - \mathbf A(t) \nonumber \\
    | \Delta \mathbf A | & = 2A \sin (\Delta \theta / 2) \nonumber
    \end{align}##

    if ##\Delta \theta \ll 1##, ##\sin (\Delta \theta / 2) \approx \Delta \theta / 2##
    ##\begin{align} | \Delta \mathbf A | & \approx 2A (\Delta \theta / 2 \nonumber \\
    & =A \Delta \theta \nonumber \\
    | \Delta \mathbf A / \Delta t | & \approx A (\Delta \theta / \Delta t) \nonumber
    \end{align}##

    if ##\Delta t \rightarrow 0##,
    ##| d \mathbf A / dt | = A (d \theta / dt)##

    But isn't it ##| d \mathbf A / dt | = 2A (d \sin (\Delta \theta / 2) / dt)##?

    Is ##v = r \omega## not accurate compared with ##v = 2r (d \sin (\Delta \theta / 2) / dt)##?
     
    Last edited: Aug 5, 2016
  2. jcsd
  3. Aug 5, 2016 #2

    BvU

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    Differentiation is exact: it is expressed as a limit and that makes for an outcome without uncertainty.

    ##| d \mathbf A / dt | = 2A (d \sin (\Delta \theta / 2) / dt)##
    On the left you have a differential quotient, but on the right you have a differential of a difference.
    With ##\theta/2 = \omega t/2## you do get the same differential ##\omega r##.
     
  4. Aug 5, 2016 #3
    By chain rule,

    ##\begin{align}
    \frac {d\sin(\theta/2)} {d(\theta /2)}\frac {d(\theta / 2)} {d\theta} & = \frac {1} {2} \cos\frac {\theta} {2} \frac {d\theta} {dt} \nonumber \\
    |\frac {d\mathbf A} {dt} | & = A\cos\frac {\theta} {2} \frac {d\theta} {dt} \nonumber
    \end{align}##

    It seems they are not the same differential.

    Also,

    ##| \frac {\Delta A} {\Delta t} | \approx A\frac {\Delta \theta} {\Delta t}##

    when ##t \rightarrow 0##, why it will become equality?
     
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