Solving Calculus Equation: a=dv/dt =>adt=dv

[Calculus 142]

Hi,
I am new to calculus, and in some books I read
a= dv/dt
=>adt=dv.

If dt means with respect to t, how is it possible multiply both sides of the equation by dt?
Is there a theorem stating this?

Thankyou.

 

[Eynstone]

adt=dv means nothing but a= dv/dt; just a prevalent abuse of notation. Also, dt is not a number which can be multiplied.

 

[Mark44]

dv and dt are differentials. For more information, see http://en.wikipedia.org/wiki/Differential_(infinitesimal ).

Differentials come up in the study of differential equations, a simple example of which is a = dv/dt. If a is a constant, we can separate this equation to dv = a dt, and integrate both sides with respect to t, to get v = at + C, where C is an arbitrary constant.

If we realize that v = ds/dt, the time rate of change of position, then we have ds/dt = at + C, which implies that ds = (at + C)dt. Integrating again with respect to t, we get s = (1/2)at^2 + Ct + D, which gives us the displacement of an object moving with a constant acceleration as a function of t.

 

[Calculus 142]

Thank you for the response.