Englishsurface integral of a vector field
Statements
![{\displaystyle \iint _{S}{\boldsymbol {F}}\cdot \mathrm {d} {\boldsymbol {\sigma }}=\iint _{D}{\boldsymbol {F}}({\boldsymbol {\Phi }}(u,v))\cdot \left({\frac {\partial {\boldsymbol {\Phi }}(u,v)}{\partial u}}\times {\frac {\partial {\boldsymbol {\Phi }}(u,v)}{\partial v}}\right)\,\mathrm {d} u\,\mathrm {d} v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53854403fb8356932f3f9b633908181029e803d6)
![{\displaystyle \iint _{S}{\boldsymbol {F}}\cdot \mathrm {d} {\boldsymbol {\sigma }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5969ece91cb9e135df70d8efbffaaf25ae5fd67b)
![{\displaystyle {\boldsymbol {F}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a728145135614a4f2c53b9332be4ea077ed10156)
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![{\displaystyle \iint _{D}f(u,v)\,\mathrm {d} u\,\mathrm {d} v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ef29995674cd146a5b2a9e2a1d053f82db6948)
![{\displaystyle {\boldsymbol {\Phi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4438f2a6c3ed8ec37856c47e37dedb3f345226f8)
![{\displaystyle \cdot }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba)
![{\displaystyle {\frac {\partial }{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48d5ed7cbe913e8e308fbe55483bd0549928db5e)
![{\displaystyle \times }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703)
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