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<Text-field style="Heading 1" layout="Normal"><Font encoding="UTF-8">integral de superf\303\255cie,</Font></Text-field>
<Text-field style="Text" layout="Normal"><Font italic="true">n.</Font> <Font bold="true">1.</Font> a integral dupla de um <Hyperlink linktarget="Dictionary:Definition/scalarfield" hyperlink="true"><Font style="Dictionary Hyperlink" family="Times New Roman">campo escalar</Font></Hyperlink><Font encoding="UTF-8">, F, no espa\303\247o Euclideano tri-dimensional real, com respeito \303\240 \303\241rea da regi\303\243o da superf\303\255cie, denotada por <Font family="MaplePi">\303\262\303\262</Font></Font> F dS; se <Font bold="true">x</Font> (<Font italic="true">u</Font>, <Font italic="true">v</Font>) para <Font italic="true">u</Font> e <Font italic="true">v</Font><Font encoding="UTF-8"> reais, \303\251 uma representa\303\247\303\243o param\303\251trica de S, ent\303\243o \303\251 igual a </Font></Text-field>
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<Font encoding="UTF-8">Em termos da parametriza\303\247\303\243o em fun\303\247\303\243o das coordenadas </Font><Font italic="true">x</Font> e <Font italic="true">y</Font>, reduz a </Text-field>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="">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</Equation>, sobre S 
<Font bold="true">b.</Font> a integral dupla de um <Hyperlink linktarget="Dictionary:Definition/scalarfield" hyperlink="true"><Font style="Dictionary Hyperlink" family="Times New Roman">campo escalar</Font></Hyperlink>, F, em <Font bold="true">R</Font><Font superscript="true">3</Font><Font encoding="UTF-8">, com respeito \303\240 \303\241rea da regi\303\243o da superf\303\255cie na dire\303\247\303\243o de um dos eixos de coordenada.  \303\211 denotada por <Font family="MaplePi">\303\262\303\262</Font></Font><Font subscript="true">S</Font>F dS<Font subscript="true" italic="true">i</Font><Font encoding="UTF-8">  e \303\251 igual a <Font family="MaplePi">\303\262\303\262</Font></Font><Font subscript="true">S</Font>F <Font italic="true">n<Font subscript="true">i</Font></Font> dS, onde <Font italic="true">n<Font subscript="true">i</Font></Font><Font encoding="UTF-8"> \303\251 a <Font italic="true">i-\303\251sima</Font> componente da normal unit\303\241ria que aponta para fora da superf\303\255cie.  </Font><Font bold="true">c.</Font><Font encoding="UTF-8"> combina\303\247\303\265es lineares de tais integrais; estas integrais de superf\303\255cie s\303\243o an\303\241logas \303\240s </Font><Hyperlink linktarget="Dictionary:Definition/curvilinearintegral" hyperlink="true"><Font style="Dictionary Hyperlink" family="Times New Roman" encoding="UTF-8">integrais curvil\303\255neas</Font></Hyperlink><Font encoding="UTF-8"> de segunda esp\303\251cie. </Font><Font bold="true">2.</Font><Font encoding="UTF-8"> menos comum, a integral dupla de uma fun\303\247\303\243o de tr\303\252s vari\303\241veis sobre uma regi\303\243o S de uma superf\303\255cie; onde F(</Font><Font italic="true">x</Font>,<Font italic="true"> y</Font>,<Font italic="true"> z</Font><Font encoding="UTF-8">) \303\251 a fun\303\247\303\243o, e </Font><Font italic="true">z </Font>= <Font italic="true">f</Font>(<Font italic="true">x, y</Font>) para <Font family="MaplePi" encoding="UTF-8">\303\241</Font><Font italic="true">x</Font>, <Font italic="true">y</Font><Font family="MaplePi" encoding="UTF-8">\303\261\303\216</Font><Font encoding="UTF-8"> D \303\251 a regi\303\243o da superf\303\255cie, a integral da superf\303\255cie 
<Font family="MaplePi">\303\262\303\262</Font></Font><Font subscript="true">S</Font>F(<Font italic="true">x</Font>,<Font italic="true"> y</Font>,<Font italic="true"> z</Font><Font encoding="UTF-8">)dx dy 
\303\251 calculada como a integral dupla 
<Font family="MaplePi">\303\262\303\262</Font></Font><Font subscript="true">D</Font>F[<Font italic="true">x</Font>,<Font italic="true"> y</Font>,<Font italic="true"> f</Font>(<Font italic="true">x, y</Font>)] dx dy<Font italic="true">.</Font></Text-field>
</Worksheet>