Calculus of Variations

This chapter is devoted to the calculus of variations which is a branch of mathematics that allows us to find a function \(y=f(x)\) that minimizes a functional—a function of function, for example, \(I=\int _a^b G(y,y',y'',x)dx\) is a functional. The calculus of variations provides answers to questions like “what is the plane curve with maximum area with a given perimeter?”. You might have correctly guessed the answer: in the absence of any restriction on the shape, the curve is a circle. But calculus of variation provides a proof and more.