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Sagot :
The flux of [tex]\vec E = -x\,\vec\imath + y\,\vec\jmath[/tex] is given by the surface integral
[tex]\displaystyle \iint_S \vec E \cdot d\vec\sigma[/tex]
where [tex]S[/tex] is the given square region, which we can parameterize by
[tex]\vec s(x, z) = x\,\vec\imath + z\,\vec k[/tex]
with [tex]0\le x\le 1[/tex] and [tex]0\le z\le 1[/tex]. The area element is
[tex]d\vec\sigma = \vec n \, dx\,dz[/tex]
where [tex]\vec n[/tex] is the normal vector to [tex]S[/tex]. Depending on the orientation of [tex]S[/tex], this vector could be
[tex]\vec n = \dfrac{\partial\vec s}{\partial x} \times \dfrac{\partial\vec s}{\partial z} = -\vec\jmath[/tex]
or [tex]-\vec n = \vec \jmath[/tex]; either way, the integral reduces to
[tex]\displaystyle \iint_S \vec E \cdot d\,\vec\sigma = \int_0^1 \int_0^1 (-x\,\vec\imath + z\,\vec k) \cdot (\pm\vec\jmath) \, dx\,dz = \boxed{0}[/tex]
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