dTdt

dT/dt, often written dTdt in plain text, is a standard notation for the instantaneous rate of change of temperature with respect to time. Here T denotes temperature, a function of time t, so dT/dt represents how quickly the temperature changes at a given moment. The concept is widely used in physics, engineering, meteorology, and related fields.

In its mathematical form, if T = T(t), then dT/dt is the derivative of T with respect to t. Its units are typically kelvin per second (K/s) or degrees Celsius per second (°C/s), though the exact units depend on how temperature and time are measured. In many practical contexts, the temperature change is described by differential equations that relate dT/dt to other quantities, such as environmental temperature or heat transfer properties.

A common example is Newton’s law of cooling, which states dT/dt = -k (T - T_env), where T_env is the ambient temperature and k is a positive constant. This model expresses the idea that the rate of cooling (or heating) is proportional to the difference between the object's temperature and its surroundings.

In systems where temperature varies across space as well as time, the full description uses a partial derivative ∂T/∂t, focusing on the local rate of change at a fixed position. Discrete data often approximate dT/dt using a finite difference: (T(t2) - T(t1)) / (t2 - t1).

dT/dt is a foundational quantity in thermodynamics, heat transfer, atmospheric science, and control theory, where the rate of temperature change informs design, analysis, and forecasting.