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Suppose there is a vector field A. Now the line integral over some curve c equals double integration of curl A over surface S enclosed by C. Now if we put the divergence theorem in this then it becomes triple integral of divergence of curl Adv over the volume enclosed by the surface. Now divergence of curl of vector A becomes zero. Now so the volume integral should be zero. So the surface and line integral is also zero. So we can do this for any line integral (where the vector is defined in the surface). So does it imply all line integrals are zero. Where am I doing the mistake? Sorry for mathjax.

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  • $\begingroup$ What are the hypotheses on using the divergence theorem? $\endgroup$ Aug 4, 2018 at 0:54
  • $\begingroup$ @JasonDeVito the vector should be valid throughout the region. The surface should enclose a volume. The region has to be simply connected. $\endgroup$
    – user187604
    Aug 4, 2018 at 0:56
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    $\begingroup$ If the surface encloses a volume, does it have a boundary curve? $\endgroup$ Aug 4, 2018 at 0:59
  • $\begingroup$ @JasonDeVito it should have a plane not a curve I guess. Except sphere or paraboloid etc. $\endgroup$
    – user187604
    Aug 4, 2018 at 1:00
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    $\begingroup$ Nope, no exceptions. If a surface encloses a volume, it has no boundary curve. So there is no way to apply the divergence theorem after applying Stokes's theorem $\endgroup$ Aug 4, 2018 at 1:20

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In order to apply the divergence theorem, your surface $S$ needs to enclose a solid. This, in particular, implies that $S$ itself has no boundary. But using Stokes's theorem relates the integral over a surface to an integral over its boundary. Thus, there is no surface $S$ for which both theorems apply.

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