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Reference.org
Surface integral
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<p> In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, particularly <a href="/facts/Multivariable_calculus/PEs5SgUF">multivariable calculus</a>, a surface integral is a generalization of <a href="/facts/Multiple_integral/Fvl3Bkam">multiple integrals</a> to <a href="/facts/Integral/V7lyV997">integration</a> over <a href="/facts/Surface_(differential_geometry)/cslVnmd6">surfaces</a>. It can be thought of as the <a href="/facts/Double_integral/Fvl3Bkam">double integral</a> analogue of the <a href="/facts/Line_integral/10s4B3Nn">line integral</a>. Given a surface, one may integrate over this surface a <a href="/facts/Scalar_field/8MtPTGob">scalar field</a> (that is, a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of position which returns a <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalar</a> as a value), or a <a href="/facts/Vector_field/ccFNuPPp">vector field</a> (that is, a function which returns a <a href="/facts/Vector_(geometric)/bCLrdksy">vector</a> as value). If a region R is not flat, then it is called a <a href="/facts/Differential_geometry_of_surfaces/cslVnmd6"><i>surface</i></a> as shown in the illustration. </p><p>Surface integrals have applications in <a href="/facts/Physics/a7aH2y08">physics</a>, particularly in the <a href="/facts/Classical_physics/gGnINVPE">classical</a> theories of <a href="/facts/Electromagnetism/oAkmALHT">electromagnetism</a> and <a href="/facts/Fluid_mechanics/NIy4tRfE">fluid mechanics</a>. </p>