Calculus {calculus of variations} studies function maxima and minima (Euler and Lagrange).
optimum
Maximum or minimum is where function has derivative equal zero: f'(y) = 0. For f'(y) = 0 = df(y) / dx, integrating gives f(y) - f(y', x) - f(y', y) * y' - f(y', y') * y'' = 0, where x = maximum or minimum domain value, y' = dy / dx, and y'' = d^2y / dx^2, with d^2 for second derivative.
If function is continuous in closed bounded domain, it has maximum and minimum.
motion through fluid
Motion through fluid takes path with least action, such as least-resistance path, brachistochrone, geodesic, and isoperimetrical curves.
iteration
Calculus of variations solves problems involving friction, inhomogeneity, and anisotropy by iterative integration. First, transform integral into sum of terms, to change boundary-value problem into initial-value problem. Then use initial value to optimize variable, using envelope of curve tangents, not point set. Repeat using previous state, to re-optimize. Solution is typically disturbance that moves through space or time.
Curved space has minimum path length {geodesic, calculus}|.
Curves {isoperimetrical curve} can bound maximum area.
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Date Modified: 2022.0225