After introducing the basic concepts of Hamiltonian mechanics in the previous chapter, we develop the modern language of vector fields, differential forms, and exterior calculus. We start by defining the vector fields through their action on functions, followed by differential one-forms by defining their actions on vector fields. We then introduce exterior products of one-forms to make two-forms and define the exterior derivative of one-forms, also producing two-forms. We also discuss the integration of forms along the surfaces in phase space. We conclude by studying one- and two-forms relevant to Hamiltonian mechanics and discuss the integral invariants of Poincaré and Poincaré-Cartan having fundamental importance for further study of canonical transformations.