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HAMJOOP
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In calculus of variation, we use Euler's equation to minimize the integral.
e.g. ∫f{y,y' ;x}dx
why we treat y and y' independent ?
e.g. ∫f{y,y' ;x}dx
why we treat y and y' independent ?
UltrafastPED said:Because there is no algebraic relation between a function and its derivative.
This is why you need boundary conditions to solve differential equations.
The calculus of variations is a branch of mathematics that deals with finding the optimal value of a function, also known as the extremum, by considering variations in the function. It is used to solve problems involving optimization, such as finding the shortest path or minimizing energy usage.
The main difference between the two is that traditional calculus deals with finding the maxima and minima of a function with a fixed set of variables, while the calculus of variations considers variations in the function itself. Traditional calculus is also concerned with finding exact solutions, whereas the calculus of variations focuses on finding the optimal solution within a certain range of values.
The calculus of variations has various applications in physics, engineering, economics, and other fields. Some examples include finding the path of least resistance in fluid dynamics, determining the shape of a hanging chain, and optimizing the shape of an airplane wing for maximum lift.
The Euler-Lagrange equation is a fundamental equation in the calculus of variations that is used to find the extremum of a functional. It relates the derivative of the function to the function itself and its derivatives. Solving this equation yields the optimal solution for the given problem.
While the calculus of variations is a powerful tool for solving optimization problems, it does have some limitations. It can only be applied to functions that have a smooth and continuous behavior, and it does not always guarantee finding the global extremum. Additionally, some problems may have no solution or an infinite number of solutions, making it difficult to determine the optimal value.