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第十五周。

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2023-06-06 22:51:29 +08:00
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18曲面积分.tex
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@@ -84,7 +84,7 @@
于是
\begin{align*}
\iint \limits_\Sigma \bvec{F} \cdot \dif \bvec{\sigma} & = \pm \iint \limits_\Delta \left(P\circ \bvec{r} A + Q\circ \bvec{r} B + R\circ \bvec{r} C\right)\dif u \dif v\\
& = \pm \iint \limits_\Delta \left(P(\bvec{r}(u, v)) \frac{\partial (y, z)}{\partial u, v} + Q(\bvec{r}(u, v)) \frac{\partial (z, x)}{\partial (u, v)} + R(\bvec{r}(u, v)) \frac{\partial (x, y)}{\partial (u, v)}\right) \dif u \dif v\\
& = \pm \iint \limits_\Delta \left(P(\bvec{r}(u, v)) \frac{\partial (y, z)}{\partial (u, v)} + Q(\bvec{r}(u, v)) \frac{\partial (z, x)}{\partial (u, v)} + R(\bvec{r}(u, v)) \frac{\partial (x, y)}{\partial (u, v)}\right) \dif u \dif v\\
& = \pm \iint \limits_\Delta \begin{vmatrix}
P \circ \bvec{r} & Q \circ \bvec{r} & R \circ \bvec{r}\\[1ex]
\dfrac{\partial x}{\partial u} & \dfrac{\partial y}{\partial u} & \dfrac{\partial z}{\partial u}\\[1em]