Note¶
Hint
\(\(\frac{\partial (x,y)}{\partial (u,v)}=\left|\begin{array}{cccc}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{array}\right|\)\)
傅里叶级数
\[
\frac{a_0}{2}+\sum_{n=1}^{+\infty}(a_n \cos \frac{n\pi x}{l}+b_n \sin \frac{n\pi x}{l})
\]
\[
\left\{
\begin{aligned}
a_n=\frac{1}{l}\int_{-l}^{l} f(x)\cos\frac{n\pi x}{l}dx, \quad n=0,1,2... \\
b_n=\frac{1}{l}\int_{-l}^{l} f(x)\sin\frac{n\pi x}{l}dx, \quad n=1,2,3...
\end{aligned}
\right.
\]
高斯公式
\[\oiint_S F\cdot dS=\iiint_V \nabla \cdot F dV\]
梯度公式
\[\oiint_S p dS=\iiint_V \nabla pdV\]
散度
\[\nabla \cdot A=\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}, A=A_x i + A_y j + A_zk\]
梯度
\[\nabla p=\frac{\partial p}{\partial x}i+\frac{\partial p}{\partial y}j+\frac{\partial p}{\partial z}k, p=p(x,y,z)\]
旋度
\[\nabla \times A=\left|\begin{array}{cccc}i & j & k \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
A_x & A_y & A_z\end{array}\right|\]
Note
\[\nabla\cdot V=\frac{1}{\partial V}\frac{D(\partial V)}{Dt}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\]
速度散度表示单位体积的运动流体微元的体积随时间的变化率
环量
\[\Gamma=-\oint_C V \cdot \mathcal{d}s\]
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