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# 曲面积分示例

## 当前任务：球体上的曲面积分。

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#### 步骤 1: 利用球体的对称性

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\begin{aligned} \iint_{\text{球体}} \Big((x-1)^2 + y^2 + z^2 \Big) \,d\Sigma = \iint_{\text{球体}} (-2x+5)\,d\Sigma \end{aligned}

#### 步骤 2: 参数化球体

\begin{aligned} &\quad \iint_{\text{球体}} (-2x+5)\,d\Sigma \\\\ &= \int_0^\pi \int_0^{2\pi} \left( -2 \underbrace{ (2\cos(t)\sin(s)) }_{\text{x 参数化的值}} +5 \right)\, \underbrace{ \left| \dfrac{\partial \vec{\textbf{v}}}{\partial t} \times \dfrac{\partial \vec{\textbf{v}}}{\partial s} \right| }_{\text{我们需要把这个算出来}} \!\!\!\!\!\! \,dt \,ds \end{aligned}

#### 步骤 3: 计算两个偏导数

\begin{aligned} \left| \dfrac{\partial \vec{\textbf{v}}}{\partial t} \times \dfrac{\partial \vec{\textbf{v}}}{\partial s} \right| \end{aligned}

\begin{aligned} \vec{\textbf{v}}(t, s) = \left[ \begin{array}{c} 2\cos(t)\sin(s) \\ 2\sin(t)\sin(s) \\ 2\cos(s) \end{array} \right] \end{aligned}
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#### 步骤 4: 计算叉乘

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#### 步骤 5: 求叉乘的大小。

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#### 步骤 6: 计算积分

\begin{aligned} &\quad \iint_{\text{球体}} f(x, y, z)\,d\Sigma \\\\ &= \iint_{\text{球体}} (-2x+5)\,d\Sigma \quad \leftarrow \text{步骤 1} \\\\ &= \int_0^\pi \int_0^{2\pi} \Big(-2(2\cos(t)\sin(s))+5\Big)\, \left| \dfrac{\partial \vec{\textbf{v}}}{\partial t} \times \dfrac{\partial \vec{\textbf{v}}}{\partial s} \right| \,dt \,ds \quad \leftarrow \text{步骤 2} \\\\ &= \int_0^\pi \int_0^{2\pi} \Big(-2(2\cos(t)\sin(s))+5\Big)\, (4\sin(s)) \,dt \,ds \quad \leftarrow \text{步骤 3, 4, 5} \\\\ &= \int_0^\pi \int_0^{2\pi} \Big(-16\cos(t)\sin^2(s)+20\sin(s) \Big) \,dt \,ds \end{aligned}

\begin{aligned} \int_0^\pi \int_0^{2\pi} \Big(-16\cos(t)\sin^2(s)+20\sin(s) \Big) \,dt \,ds = \end{aligned}