Lagrange's Equation
The partial differential equation
| (1+f_y^2)f_(xx)-2f_xf_yf_(xy)+(1+f_x^2)f_(yy)=0 |
(Gray 1997, p. 399), whose solutions are called minimal surfaces. This corresponds to the mean curvature H equalling 0 over the surface.
| y=xf(y^')+g(y^') |
is sometimes also known as Lagrange's equation (Zwillinger 1997, pp. 120 and 265-268).
See also
d'Alembert's Equation, Mean Curvature, Minimal SurfaceExplore with Wolfram|Alpha
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References
do Carmo, M. P. "Minimal Surfaces." §3.5 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 41-43, 1986.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.Zwillinger, D. "Lagrange's Equation." §II.A.69 in Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 120 and 265-268, 1997.Referenced on Wolfram|Alpha
Lagrange's EquationCite this as:
Weisstein, Eric W. "Lagrange's Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LagrangesEquation.html