# How to derive $dW = dE$? [closed]

From the definition of power:

$$P \ = \ \frac{dW}{dt} \ = \ \frac{dE}{dt}$$

Multiplying both sides by $$dt$$, we get:

$$dW = dE$$

$$1.$$ What does this imply?

$$2.$$ Are there any other (perhaps more rigorous) way of proving this?

• It implies that you are equating $W$ and $E$ up to a function that has no time dependence. What is $W$ and what is $E$ here? – Aaron Stevens Jul 19 '19 at 0:13
• This question scratches the tip of a very large iceberg. Search here for questions on work and energy. – garyp Jul 19 '19 at 0:25
• If you ask what $dW = dE$ imeans mathematically: it means $W = E +c$ where $c$ is a constant. – Fabian Jul 19 '19 at 2:58
• Work is transferred energy. That's about it: $W=\Delta E$ means that the work $W$ done by a force equals the amount of energy $\Delta E$ that's moved from one (sub)system to another in the interaction. When you write that with differentials, you get $dW=dE$. – stafusa Jul 20 '19 at 8:11
• @Fabian I can understand that relationship by either intuition or integrating both sides. But I have one contradictory idea in my mind that $W = E + C$ implies that W and E are both the indefinite integrals of their derivative. As far as I know indefinite integral of a force is energy, and the definite integral is work. How can work, a definite integral, be in the relation of indefinite integral ??? – curious Sep 29 '19 at 8:07

$$\Delta U = Q + W$$ (internal energy)
$$\Delta E = \Delta E_m + Q + W$$ (total energy)