Calculus of Variations

What is the shortest curve that connects two points? What is the fastest path that a particle can travel from A to B by the force of gravity? Given two fixed points, A and B above the x-axis, which curve between the two points can generate the minimal surface area by rotating it about the x-axis? Those problems share a common attribute, which is to discover the curve that can produce an extreme result to a problem. To find the point of extreme values in the fundamental calculus, we set the function's derivative to zero and solve for answers. The calculus of variations applies similar ideas to find the function (the curve) that produces an extreme value. The thesis aims to discuss the origin of the technique of finding the "best" curves and how to apply the technique to find the solutions to the problems mentioned above.