Differential equations are mathematical equations that involve functions and their derivatives, representing physical phenomena and changes in various fields such as physics, engineering, and economics. They are essential for modeling and solving problems where quantities change continuously, providing insights into the behavior and dynamics of complex systems.
Multivariable calculus extends the principles of single-variable calculus to functions of multiple variables, allowing for the analysis and optimization of systems with more than one input. It is essential for understanding complex phenomena in fields such as physics, engineering, economics, and beyond, where interactions between multiple varying quantities need to be quantified and optimized.
Smoothness of functions refers to the degree to which a function is differentiable and how its derivatives behave. It is crucial in understanding the behavior of functions, impacting areas such as approximation theory, numerical analysis, and the solution of differential equations.