Mathematical physics serves as the foundation for expressing and solving physical problems using rigorous mathematical structures and techniques. The discipline of mathematical physics bridges the gap between abstract mathematical theories and physical phenomena, providing a formal framework through which the laws of nature are defined, analyzed, and interpreted. The foundations of mathematical physics rest on several key mathematical structures, including calculus, linear algebra, differential equations, and complex analysis. These mathematical tools are essential for formulating physical theories, predicting experimental outcomes, and uncovering deeper insights into the nature of the physical world.

One of the fundamental mathematical tools in physics is calculus, which provides the framework for describing the motion of objects, the behavior of fields, and the change of physical quantities over time and space. Newton and Leibniz independently developed the principles of differential and integral calculus, which allowed for the precise formulation of classical mechanics and gravitation. Differential calculus deals with rates of change and slopes of curves, which are essential in defining velocity, acceleration, and force. Integral calculus, on the other hand, is used to calculate areas under curves and accumulated quantities, such as work and potential energy.