The variation of a derivative, also known as calculus of variations, is a mathematical theory that deals with finding the optimal path or function that minimizes or maximizes a given functional. It involves finding the function that satisfies certain conditions and produces the most extreme value for a given functional.
Calculus of variations has numerous applications in physics, engineering, economics, and other fields. Some examples include finding the shortest distance between two points, determining the shape of a hanging chain, optimizing the shape of a wing for maximum lift, and finding the path of a projectile with the least air resistance.
Standard calculus deals with finding the maximum or minimum values of a function with respect to a variable. In contrast, calculus of variations deals with finding the function itself that produces the most extreme value for a given functional. It involves considering an entire family of functions rather than a single one.
The basic principles of calculus of variations include the Euler-Lagrange equation, which is used to find the extremum of a functional, and the fundamental lemma of the calculus of variations, which states that if a function satisfies the Euler-Lagrange equation, it is a critical point of the functional. Other principles include the method of variation of parameters and the principle of least action.
While calculus of variations may seem intimidating at first, it can be easily understood with a strong foundation in standard calculus and some practice. It is a fundamental concept in mathematics and has numerous real-world applications. With patience and perseverance, anyone can grasp the principles and applications of calculus of variations.