Aeronautics and astronautics rely heavily on advanced mathematics to model and analyze the behavior of air and spacecraft systems. Key mathematical tools include calculus for understanding rates of change in motion and flow, linear algebra for solving systems of equations in control and structural analysis, and differential equations for modeling dynamics, vibrations, and orbital mechanics. Vector calculus is critical for fluid flow, force, and field analysis, while complex analysis and Fourier transforms are used in signal processing and aerodynamic studies. Additionally, probability and statistics help in reliability assessment and risk analysis of aerospace systems.