Integral of surface

In Mathematical, an integral of surface is a Intégrale definite on a whole surface which can be space curve. For a given surface, one can integrate on a scalar Champ or a vector Field.

The integrals of surface have many applications in the classical theory of the electromagnetism.

Integral of surface on a scalar field

To find a formula explicit of the integral of surface, it is generally necessary to parameterize surface S in question by considering a curvilinear Frame of reference, like the Longitude and the Latitude on a Sphère. Once the parameter setting X (S, T) found, where S and T varies in an area of the plan, the integral of surface of a scalar field is given by:

$\backslash \; int\_S\; F\; \backslash \; mathrm\; dS\; =\; \backslash \; iint\_T\; F\; \backslash \; bigl\; (\backslash \; mathbf\; \{X\}\; (S,\; T)\; \backslash \; bigr)\; \backslash \; left\; \backslash |\backslash \; frac\; \{\backslash \; partial\; \backslash \; mathbf\; \{X\}\}\; \{\backslash \; partial\; S\}\; \backslash \; wedge\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; mathbf\; \{X\}\}\; \{\backslash \; partial\; T\}\; \backslash \; right\; \backslash |\backslash ;\; \backslash \; mathrm\; ds\; \backslash ,\; \backslash \; mathrm\; dt$

Moreover the surface of S is given by:

$\backslash int\_S\; \backslash mathrm\; dS$.

See too

• Integral
• Integral curvilinear
• Theorem of flow-divergence
• Theorem of Stokes
• Pseudovecteur

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