For two-independent-variable functions, calculate integral {double integral} by holding first variable constant and integrating over second variable, then holding second variable constant and integrating over first variable, and then adding results. It does not matter which variable is first. Double-integral domain can be surface {closed region} inside closed curve.
To find surface area {area, integration}, take double integral over differential area (1 + (df(x,y) / dx)^2 + (df(x,y) / dy)^2)^0.5 * dx * dy.
Triple integral, over volume, of scalar product of del operator and vector function equals double integral, over surface, of scalar product of function and normal vector to surface {divergence theorem}.
If two solids have equal altitudes and all sections parallel to base have same ratio, volumes have same ratio {Cavalieri's theorem} {Cavalieri theorem}.
The gradient of scalar electric potential is vector electric field. A scalar function has a divergence of the gradient {Laplacian}: D^2f/Dx^2 + D^2f/Dy^2 + D^2f/Dz^2, where D is partial derivative. Potential relates to charge density as Laplacian of potential equals negative of charge density divided by electrostatic constant (Poisson's equation). If charge density is zero, Laplacian of potential equals zero (Laplace's equation).
Curved-surface area is not the limit of surface's plane-triangle areas {Schwarz's paradox} {Schwarz paradox}.
In a plane region, if function has no singularities, is single-valued, and solves the potential equation, double integral of ((du/dx)^2 + (du/dy)^2) * dx * dy over x and y has a minimum {Dirichlet principle} {Thomson principle}.
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Date Modified: 2022.0225