First do a multivariate function and function analogy:

For multivariate function f(x 1, x2, ..., xn), its independent variable is an n-dimensional array (x 1, x2, ..., xn);

For the functional F=F(y), figuratively speaking, its independent variables can correspond to a function curve y=y(x), because there are infinitely many points on the curve, and the y coordinate of each point is the independent variable of the functional, then the infinitely many points on the curve correspond to infinitely many independent variables, and the functional f is the function of the infinitely many independent variables.

For multivariate functions, it has total differential df=? f/？ x 1*dx 1+？ f/？ x2*dx2+...+? f/？ Xn*dxn, that is, the change of function value when each independent variable changes slightly.

For functional f, its total differential is variation, that is, when the Y coordinate of each point on the curve line changes slightly, the function value of the whole functional changes.

For example:

A common example is the steepest descent line problem. When an object moves from one point A to another point B in space under the action of gravity, it can pass through an infinite number of trajectories, and each trajectory corresponds to a different movement time t, so t is the functional of these trajectories. The minimum trajectory change of t is 0, that is, when an object takes a trajectory close to it, the change of required time t approaches 0.