dT2dt

dT2dt is a textual shorthand for the time derivative of the square of a quantity T with respect to time t. The conventional notation is d(T^2)/dt, and it assumes that T is a differentiable function of t.

For a scalar T(t), the chain rule gives d/dt (T^2) = 2 T dT/dt. This is because the derivative of T^2 with respect to t multiplies the rate of change of T by twice the current value of T. If T is constant, d/dt (T^2) = 0.

If T is a vector-valued function T(t), one often considers the squared magnitude, ||T||^2, rather than T^2. In that case, d/dt (||T||^2) = 2 T · dT/dt, where the dot denotes the dot product. This generalizes the scalar result to the common interpretation of the square of a vector’s length.

Examples help illustrate the idea. If T(t) = t, then d/dt (T^2) = d/dt (t^2) = 2t. If T(t) = t^2, then d/dt (T^2) = d/dt (t^4) = 4t^3. If T(t) = sin t (scalar), then d/dt (T^2) = 2 sin t cos t = sin 2t.

In practice, d(T^2)/dt appears in differential equations and dynamical models where the rate of change of a squared quantity is relevant, or as part of energy-like expressions involving T. Interpretations depend on the specific meaning of T in the given context.