1. PF Insights is off to a great start! Fresh and interesting articles on all things science and math. Here: PF Insights

A= dv/dt

  1. I see in many text that a = dv/dt implies that

    dv = a dt

    How is that possible, can anybody please explain me. As far as i know dv/dt is a symbol for derivative of v w.r.t t and not ratio between dv and dt.
     
  2. jcsd
  3. What you wrote is in differential form.
     
  4. jgens

    jgens 1,621
    Gold Member

  5. dv/dt is not a symbol. It is the mathematical formula for the derivative of v with respect to t. This is how one would show any derivative of a dependent variable with respect to an independent variable.
     
  6. And what did I say :-)
     
  7. arildno

    arildno 12,015
    Science Advisor
    Homework Helper
    Gold Member

    Now, WHY can we utilize at times the dv=adt formula, in particular, WHERE is it usable?

    Answer:

    When doing integration with the technique called substitution of variables (i.e the "inverse" of the chain rule):

    Given a=dv/dt, we have, trivially:
    [tex]\int_{t_{1}}^{t_{2}}adt=\int_{t_{1}}^{t_{2}}\frac{dv}{dt}dt[/tex]
    But the right-hand side can, by the theorem of substitution of variables, be reformulated, giving the identity:
    [tex]\int_{t_{1}}^{t_{2}}adt=\int_{v(t_{1})}^{v(t_{2})}dv=\int_{v_{1}}^{v_{2}}dv[/tex]

    Now, by IGNORING that the limits of integration actually refers to the limits of DIFFERENT variables, we "may say" that the "integrands" are equal, i.e, adt=dv!


    Thus, adt=dv should, at this stage of your education, be regarded as notational garnish (or garbage, if you like!)
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?