Note 1 to entry: For a surface defined by r = f(u, v) where (u,v)∈U⊆R2, the scalar surface element is given by
|∂f∂u×∂f∂v|⋅dudv.
Note 2 to entry: For a surface defined by the equation z = f(x, y), the scalar surface element is given by
√1+(∂f∂x)2+(∂f∂y)2dxdy.
Note 1 to entry: For a surface defined by r = f(u, v) where (u,v)∈U⊆ R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaacIcacaWG1bGaai ilaiaadAhacaGGPaGaeyicI4CcLbuacaqGvbGccqGHgksZieqacaWF sbWaaWbaaSqabeaajugabiaabkdaaaaaaa@440F@ , the scalar surface element is given by
| ∂f ∂u × ∂f ∂v |⋅dudv MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaWaaSaaae aacqGHciITcaWHMbaabaGaeyOaIyRaamyDaaaacqGHxdaTdaWcaaqa aiabgkGi2kaahAgaaeaacqGHciITcaWG2baaaaGaay5bSlaawIa7ai abgwSixlGacsgacaWG1bGaciizaiaadAhaaaa@4A27@ .
1+ ( ∂f ∂x ) 2 + ( ∂f ∂y ) 2 dxdy MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaakaaabaGaaGymai abgUcaRmaabmaabaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRa amiEaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRa WkdaqadaqaamaalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kaadMha aaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaqabaGcciGGKb GaamiEaiGacsgacaWG5baaaa@4A2F@ .