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  "body": "\n## Summary\n\nThe calculus of variations studies the extreme values of functionals, that is, of maps that send functions to numbers.\nExamples of functionals are length, area and energy.\nA typical functional is an integral $\\int_\\Omega L(x, u, \\nabla u) \\, dx$, minimised among functions $u$ with given boundary values.\nIts smooth critical points solve a partial differential equation, the [[Euler–Lagrange equation|PS0VV149ET]].\nConversely, many equations of physics and geometry are Euler–Lagrange equations, and the calculus of variations gives existence through the [[direct method|TE379HBTY0]].\n\n## Explanations\n\n### L1 human\n\nOrdinary calculus finds the point where a function of a number is smallest.\nThe calculus of variations finds the curve, surface or function that makes a quantity smallest.\nExamples are the shortest path between two points, the soap film of least area spanning a wire, and the shape of a hanging chain of least potential energy.\nFor the shortest path in the plane, the functional is the length $\\int_a^b \\sqrt{1 + y'(x)^2} \\, dx$, and the answer is the straight line.\n",
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