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

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Discussion Overview

The discussion revolves around the manipulation of the equation a = dv/dt in calculus, specifically addressing the validity of multiplying both sides by dt and the implications of such operations. The scope includes conceptual understanding of differentials and their application in calculus, particularly in the context of differential equations.

Discussion Character

  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant questions the legitimacy of multiplying both sides of the equation a = dv/dt by dt, seeking clarification on whether there is a theorem that supports this operation.
  • Another participant asserts that the expression adt = dv is an abuse of notation and emphasizes that dt is not a number that can be multiplied in this context.
  • A different participant explains that dv and dt are differentials and provides a brief derivation showing how to separate the equation to integrate and find the velocity and displacement of an object under constant acceleration.

Areas of Agreement / Disagreement

Participants express differing views on the manipulation of the equation and the nature of differentials. There is no consensus on the validity of multiplying both sides by dt, and the discussion remains unresolved regarding the interpretation of these mathematical operations.

Contextual Notes

Some limitations include the potential misunderstanding of differentials and the notation used in calculus. The discussion does not resolve the mathematical steps involved in the manipulation of the equation.

Calculus 142
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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.
 
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adt=dv means nothing but a= dv/dt; just a prevalent abuse of notation. Also, dt is not a number which can be multiplied.
 


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.
 
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Thank you for the response.
 

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