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Use Stokes' Theorem to evaluate $$\iint_{S} \text {curl} \mathbf{F} \cdot d \mathbf{S} \cdot \mathbf{F}(x, y, z)=z e^{y} \mathbf{i}+x \cos y \mathbf{j}+x z \sin y \mathbf{k},$$
$$S$$ is the hemisphere $$x^{2}+y^{2}+z^{2}=16, y \geq 0,$$ oriented in the direction of the positive $$y$$ -axis

$$x^{2}+y^{2}+z^{2}=16 \quad y \geq 0$$

$$y=0 \quad x^{2}+z^{2}=16$$


$$\left\{\begin{array}{l}{x=4 \cos (-t)} \\ {z=4 \sin (-t)}\end{array}\right.$$

$$\mathbf{F}(x, y, z)=z e^{y} \mathbf{i}+x \cos y \mathbf{j}+x z \sin y \mathbf{k} $$

$$f(x, y, z)=z \hat{i}+x \hat{j} $$

$$\iint_{s} \text {curl} f d s=\int_{c} f(r(t)) r^{\prime}(t) d t$$

$$r(t)=4 \cos (-t) \hat{i}+4 \sin (-t) \hat{k} $$

$$r^{\prime} (t)=-4 \sin t \hat{i}+(-4 \cos t) \hat{k}$$

$$f(r(t))=4 \sin (-t) \hat{i}+4 \cos (-t) \hat{j} $$

$$\int_{0}^{2 \pi}\left[4 \sin (-t)\hat{i}-4 \cos(-t) \hat{j}\right]\left [-4 \sin t \hat{i}-4 \cos t \hat{k}\right]$$

$$=\int_{0}^{2 \pi} 16 \sin ^{2}(t)d t=16 \pi$$

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