# A= dv/dt

1. Aug 4, 2009

### vijay_singh

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. Aug 4, 2009

### Cyrus

What you wrote is in differential form.

3. Aug 4, 2009

4. Aug 4, 2009

### bucher

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.

5. Aug 4, 2009

### vijay_singh

And what did I say :-)

6. Aug 4, 2009

7. Aug 4, 2009

### arildno

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

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:
$$\int_{t_{1}}^{t_{2}}adt=\int_{t_{1}}^{t_{2}}\frac{dv}{dt}dt$$
But the right-hand side can, by the theorem of substitution of variables, be reformulated, giving the identity:
$$\int_{t_{1}}^{t_{2}}adt=\int_{v(t_{1})}^{v(t_{2})}dv=\int_{v_{1}}^{v_{2}}dv$$

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!)