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| − | A formula that expresses the connection between the flow of a vector field through a two-dimensional oriented manifold and the circulation of this field along the correspondingly oriented boundary of this manifold. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s0903101.png" /> be an oriented piecewise-smooth surface, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s0903102.png" /> be the unit normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s0903103.png" /> (at those points, of course, where it exists), which defines the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s0903104.png" />, and let the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s0903105.png" /> consist of a finite number of piecewise-smooth contours. The boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s0903106.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s0903107.png" />, and is oriented by means of the unit tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s0903108.png" />, such that the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s0903109.png" /> obtained is compatible with the orientation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031011.png" />.
| + | #REDIRECT[[Stokes theorem]] |
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| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031012.png" /> is a continuously-differentiable [[Vector field|vector field]] in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031013.png" />, then
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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| − | (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031015.png" /> is the area element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031017.png" /> is the differential of the arc length of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031019.png" />) or, in coordinate form,
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031020.png" /></td> </tr></table>
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| − | Stated by G. Stokes (1854).
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| − | Stokes' formula is also the name given to a generalization of formula , which represents the equality between the integral of the exterior differential of a [[Differential form|differential form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031021.png" /> over an oriented compact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031022.png" /> and the integral of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031023.png" /> itself along the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031025.png" /> (the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031026.png" /> is taken to be compatible with that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031027.png" />): | |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090310/s09031028.png" /></td> </tr></table>
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| − | Other particular cases of this formula are the [[Newton–Leibniz formula|Newton–Leibniz formula]], the [[Green formulas|Green formulas]] and the [[Ostrogradski formula|Ostrogradski formula]].
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| − | ====Comments====
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| − | ====References====
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian) {{MR|}} {{ZBL|0692.70003}} {{ZBL|0572.70001}} {{ZBL|0647.70001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Spivak, "Calculus on manifolds" , Benjamin (1965) {{MR|0209411}} {{ZBL|0141.05403}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> C. deWitt-Morette, "Analysis, manifolds, physics" , North-Holland (1977) pp. 205 (Translated from French) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Triebel, "Analysis and mathematical physics" , Reidel (1986) pp. 375 {{MR|0914975}} {{MR|0880867}} {{ZBL|0607.46047}} </TD></TR></table>
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