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Question 1.58: Check Stokes’ theorem for the function v = y ˆz, using the t...

Check Stokes’ theorem for the function v = y \hat{z} , using the triangular surface shown in Fig. 1.51. [Answer: a^2]

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v · dl = y dz.

(1)   Left side: z = a − x; dz = −dx; y = 0. Therefore ∫v · dl = 0.

(2)   Bottom: dz = 0. Therefore  ∫v · dl = 0.

(3)   Back: z = a −\frac{1}{2} y;  dz = −1/2  dy;   y : 2a\rightarrow 0.  \int{}v · dl =\int\limits_{2a}^{0}{}y\left(-\frac{1}{2}dy \right)=-\frac{1}{2} \frac{y^2}{2}\mid ^{0}_{2a} =\frac{4a^2}{4}=a^2 .

Meanwhile, ∇\times v=\hat{x},  so  \int{}\left(∇\times v\right) · da is the projection of this surface on the xy plane =\frac{1}{2}\cdot a\cdot 2a=a^2 .

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