Flow (mathematical)

In vectorial Analysis with three dimensions, one calls flow of a vector Field $\backslash \; vec\; \{E\}$ through a Surface directed $\backslash \; Sigma$ the scalar quantity

$\backslash \; Phi\; \backslash \; equiv\; \backslash \; iint\_\; \{\backslash \; Sigma\}\; \backslash \; vec\; \{E\}\; \backslash \; cdot\; \{\backslash \; rm\; D\}\; \backslash \; vec\; \{S\}$

If $\backslash \; Sigma$ is a closed surface (one says also without edge ) surrounding a volume $\backslash \; mathcal\; \{V\}$ then flow can be given in another manner, by calling upon the Théorème of flow-divergence:

$\backslash \; Phi=\; \backslash \; oint\_\; \{\backslash \; Sigma\}\; \backslash \; vec\; \{E\}\; \backslash \; cdot\; \{\backslash \; rm\; D\}\; \backslash \; vec\; \{S\}\; =\; \backslash \; iiint\_\; \{\backslash \; mathcal\; \{V\}\}\; \backslash \; mathrm\; \{div\}\; \backslash \; vec\; \{E\}\; \backslash \; \{\backslash \; rm\; D\}\; ^3x$

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