A proof that torque is equal to the time-derivative of angular momentum can be stated as follows: The concept of torque in physics, also called moment or couple, originated with the work of Archimedes on levers. ... In mathematics, the derivative is one of the two central concepts of calculus. ... In physics, angular momentum is analogous to (linear) momentum except that it applies to rotating objects. ...

The definition of angular momentum for a single particle is:

where "×" indicates the vector cross product. The time-derivative of this is: In mathematics, the cross product is a binary operation on vectors in a three dimensional vector space. ...

This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definitions of velocity v = dr/dt, acceleration a = dv/dt and linear momentum p = mv, we can see that: In mathematics, the product rule of calculus, which is also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... // Basic explanation The velocity of an object is simply its speed in a particular direction. ... Acceleration is the time rate of change of velocity, and at any point on a v-t graph, it is given by the gradient of the tangent to that point In physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. ... In physics, momentum is a physical quantity related to the velocity and mass of an object. ...

But the cross product of any vector with itself is zero, so the second term vanishes. Hence with the definition of force F = ma, we obtain: In physics, a force is an external cause responsible for any change of a physical system. ...

And by definition, torque τ = r×F. Note that there is a hidden assumption that mass is constant — this is quite valid in non-relativistic mechanics. Also, total (summed) forces and torques have been used — it perhaps would have been more rigorous to write: