Verify the Divergence theorem for the vector field A(x, y) = αrr + βzz where V and S are the volume and surface of the cylinder shown in Fig. A.8. Here, α and β are constants.

Question

Verify the Divergence theorem for the vector field A(x, y) = [latex]αr\hat{r} + βz\hat{z}[/latex] where V and S are the volume and surface of the cylinder shown in Fig. A.8. Here, α and β are constants.

Accepted Answer

We have already evaluated [latex]\oint_{s} A· \hat{n}[/latex] ds in Example A.4.3. Specifically, we have

[latex]\int_{s} A ⋅ ds = βL πa^{2} + αa^{2}2πL[/latex]

= (2α + β)π[latex]a^{2}[/latex]L. (A.46)

It remains to evaluate [latex]\int_{v}[/latex] ∇ · A dv. We use cylindrical coordinates and obtain

∇ · A = [latex]\frac{1}{r}\frac{∂(αr^{2})}{∂r} + \frac{∂(βz)}{∂z}[/latex]

= 2α + β.

Therefore,

[latex]\int_{v} ∇ · A dv = (2α + β)\int_{v} dv[/latex]

= (2α + β)π[latex]a^{2}[/latex]L,

which is the same as Eq. (A.46).

Question A.5.2: Verify the Divergence theorem for the vector field A(x, y) =...

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