Question 27.18: Verify the divergence theorem for the vector field v = x²i +......

Engineering Mathematics A foundation for Electronic, Electrical, Communications and Systems Engineers [1040491]

Verify the divergence theorem for the vector field $v=x^{2} i + \frac{1}{2}y^{2} j + \frac{1}{2} z^{2}k$ over the unit cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1.

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Firstly we need to evaluate $\textstyle\oint_{S}\mathbf{V}\bullet dS$ where S is the surface of the cube. This integral has been calculated in Example 27.17 and shown to be 2.

Secondly we need to calculate $\int _v$ div v dV over the volume of the cube. This has been done in Example 27.16 and again the result is 2.

We have verified the divergence theorem that $\oint_{S}\mathbf{v}\cdot\mathrm{d}\mathbf{S}=\int_{V}\mathrm{d}\mathbf{i}\mathbf{v} \mathbf{v}\mathrm{d}V.$

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