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    <title>Gauss' Theorem</title>
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        <div class="title">Gauss' Theorem</div>

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            <p>
                Let \( \mathbf{u} \) be a continuously differentiable vector field, defined in a volume \( V \).
                Let \( S \) be the closed surface forming the boundary of \( V \) and let \( \mathbf{n} \) be the unit outward normal to \( S \). Then the Gauss’ theorem (divergence theorem) states that

                <img src="Figures/Lecture1A_DivergenceTheorem.png" alt="DivergenceTheorem" style="max-width: 29%; height: auto; display: block; margin: auto;">
            </p>
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        <div class="description">
            <p>Roughly speaking, the divergence theorem states that the
                total amount of expansion of \( \mathbf{u} \) within the volume \( V \) is equal to the flux of \( \mathbf{u} \) out of the surface \( S \)
            </p>
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        <p>
            <span class="sql">Proof</span>
        </p>
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            <img src="Figures/Lecture1A_SmallSubvolumesDivergenceTheorem.png" alt="SmallSubvolumesDivergenceTheorem" style="max-width: 39%; height: auto; display: block; margin: auto;">
            <p>The volume \( V \) is divided into a large number of small subvolumes \( \delta V_i \) with surfaces \( \delta S_i \).
                The proof of the divergence theorem then follows naturally from the physical definition of
                the divergence in terms of a surface integral <img src="Figures/Lecture1A_DivergenceVectorField.png"
                alt="DivergenceVectorField"
                style="height: 2.5em; vertical-align: middle;"><br>
                Within each of the subvolumes, \( \nabla \cdot \mathbf{u} \) is defined by <img src="Figures/Lecture1A_DivergenceNablaDotu.png"
                alt="DivergenceNablaDotu"
                style="height: 2.5em; vertical-align: middle;"> where the approximation becomes exact in the limit \( \delta V_i \to 0 \)</p>
            </div>

        <p>Now multiply both sides of <img src="Figures/Lecture1A_DivergenceNablaDotu.png"
            alt="DivergenceNablaDotu"
            style="height: 2.5em; vertical-align: middle;"> by \( \delta V_i \) and add the contributions
            from all the subvolumes: <img src="Figures/Lecture1A_ContributionAllSubvolumes.png"
            alt="ContributionAllSubvolumes"
            style="height: 2.5em; vertical-align: middle;">
        </p>


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            <p class="footnote">
                <sup>1</sup>P. C. Matthews, Vector Calculus, New York:Springer-Verlag, 1998.
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