I don't understand your assumption. ##r=r_+## and ##r=r_{++}## are two differents points such that ##r_{++}>r_+##WWGD said:Can you define #r_{+} , r_{++}## for us? Do it through a forehand or backhand, either is OK ;).
Sorry, a tennis joke; guess you're tired of them ;).PeteSampras said:I don't understand your assumption. ##r=r_+## and ##r=r_{++}## are two differents points such that ##r_{++}>r_+##
Variational calculus is a branch of mathematics that deals with finding the optimal value of a functional, which is a mathematical expression that takes in a function as its input and outputs a real number. It is commonly used in physics and engineering to solve problems involving optimization, such as finding the path of least resistance or the shape of a curve that minimizes a certain property.
In variational calculus, dg(r=r+) and dg(r=r++) refer to the derivatives of a functional with respect to a function. The difference between the two is that dg(r=r+) is the derivative at a specific point r, while dg(r=r++) is the derivative at a point infinitesimally close to r. In other words, dg(r=r++) is the limit of dg(r=r+) as r approaches r+.
In general, dg(r=r+) and dg(r=r++) will be equal if the function is continuous and differentiable at r. However, if the function has a discontinuity or a sharp change at r, then the derivatives will not be equal. This is because the limit of dg(r=r+) as r approaches r+ will not exist in this case.
Knowing when dg(r=r+) ≠ dg(r=r++) is important because it can affect the solutions obtained from variational calculus. If the derivatives are not equal, then the optimal value of the functional may be different depending on whether we use dg(r=r+) or dg(r=r++) in our calculations. It is important to be aware of this possibility and consider both cases when solving problems with variational calculus.
The best way to determine when dg(r=r+) ≠ dg(r=r++) is to analyze the function and its derivatives at the point r. If there is a discontinuity or a sharp change at r, then the derivatives will not be equal. Additionally, we can also use the definition of the limit to check if the limit of dg(r=r+) as r approaches r+ exists. If it does not, then dg(r=r+) ≠ dg(r=r++) at that point.