This definition of Legendre transformation is the one originally introduced by Legendre in his work in 1787, and still applied by physicists nowadays. Indeed, this definition can be mathematically rigorous if we treat all the variables and functions defined above, e.g. f , x 1 , ⋯ ⋯ , x n , p 1 , ⋯ ⋯ , p n , as differentiable functions defined on an open set of R n or on a differentiable manifold, and d f , d x i , d p i their differentials (which are treated as cotangent vector field in the context of differentiable manifold). And this definition is equivalent to the modern mathematicians' definition as long as f is differentiable and convex for the variables x 1 , x 2 , ⋯ ⋯ , x n .