Field equations are mathematical expressions that describe how physical fields, such as electromagnetic or gravitational fields, behave and interact with matter. They are fundamental in physics, providing a framework for understanding forces and energy in various contexts. For example, the Maxwell equations govern the behavior of electric and magnetic fields, while Einstein's field equations relate the geometry of spacetime to the distribution of mass and energy. These equations often take the form of partial differential equations, which involve multiple variables and their rates of change. Solving field equations allows scientists to predict how fields evolve over time and space, leading to insights in areas like classical mechanics, quantum physics, and general relativity.