Abstract: The Legendre transform is an important tool in theoretical physics, which plays a critical role in classical mechanics, statistical mechanics, and thermodynamics. Yet, in undergraduate courses, the power of mo- tivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by the Fourier trans- form. We modify the formal mathematics of the Legendre transform of multivariate function, resulting in manifestly symmetric equations, deriving the relation between the Legendre transform and the partial differentiation. Then we bring the results in the classical mechanics, derive canonical equation, and bring the results in the thermodynamic, as well as thermodynamic functions and Maxwell relation are also derived.

Key words: Legendre transformation, symmetry, classical mechanics, thermodynamics