Surface integral

A surface integral accumulates a scalar quantity over a surface or measures a vector field’s flux through it. Its value depends on the surface’s geometry and, for flux, its orientation.

Surface area element: An infinitesimal area measure on a surface, often written using a parameterization or local coordinates. It supplies the area weighting used to add contributions across a surface.
Multiple integral: An integral over a region in two or more dimensions. Surface integrals are often evaluated as double integrals over parameter domains.
Mass of a surface: The total mass of a thin surface with a specified mass density per unit area. Integrating surface density gives the mass of a thin sheet or membrane.
Line integral: An integral that accumulates a scalar or vector quantity along a curve. It integrates along a one-dimensional path rather than across a surface.
Parametric surface: A surface described by a map from a region in the plane into three-dimensional space. A parameterization turns a surface integral into an ordinary double integral.
Scalar field: A function assigning a scalar value to each point in a space or region. A scalar surface integral adds field values weighted by surface area.
Center of mass: The position representing the mass-weighted average location of a body or system. Surface integrals of density and position determine a thin body's center of mass.
Volume integral: An integral that accumulates a quantity throughout a three-dimensional region. It covers a volume rather than a two-dimensional surface.
Surface orientation: A consistent choice of direction for the normal vectors to an oriented surface. Flux integrals change sign when the chosen orientation is reversed.
Vector field: A function assigning a vector to each point in a space or region. A vector surface integral measures how the field crosses the surface.

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