How did Jean Le Rond d’Alembert contribute to the development of partial differential equations?

Jean Le Rond d’Alembert was a pioneer in the development of partial differential equations, a branch of calculus that he helped establish through his rigorous mathematical research.1

His foundational contributions include:

• Initial Publications: He published his first papers on the subject in his 1747 work, Réflexions sur la cause générale des vents, which earned him a prize from the Berlin Academy.1
• Vibrating Strings: In 1747, he applied his new calculus to the complex problem of vibrating strings in his Recherches sur les cordes vibrantes, a landmark study in the application of partial differential equations to physical phenomena.1
• Fluid Dynamics and Mechanics: Building on his earlier work regarding the equilibrium and motion of fluids, d’Alembert utilized these mathematical methods to analyze the motion of bodies of various shapes and to investigate the resistance of fluids.1

Through these efforts, d’Alembert transformed the study of calculus, providing essential tools for modeling physical systems that remain significant in mathematical science today.1