dWdt

dWdt is the notation used in physics and engineering for the instantaneous rate of change of work with respect to time, commonly identified with instantaneous power. If W(t) denotes the work done on a system up to time t, then P(t) = dW/dt is the power delivered at that moment. Work is defined by dW = F · dr, where F is the force acting on a body and dr is an infinitesimal displacement. When the point of application moves with velocity v, dW/dt = F · v. Consequently, for a single force on a particle, P = F · v, and for rotational motion, P = τ · ω, where τ is torque and ω is angular velocity. The units of dW/dt are joules per second, or watts (W).

The relationship between work and power is integral: W(t) = ∫ P(t) dt + W(0). The work-energy theorem connects total work to changes in kinetic energy: ΔK equals the net work done. Sign conventions are important: positive P indicates energy is being transferred to the system, while negative P indicates energy is being extracted.

In other domains, the concept translates similarly. In electrical engineering, instantaneous power is P = V I, the product of voltage and current. The idea of dW/dt as a rate of energy transfer appears across mechanics, thermodynamics, and electromechanics, serving as a fundamental link between forces, motion, and energy flow. See also work, power, work-energy theorem, torque, and energy transfer.