多元函数积分学

向量的运算

$$(a_x,a_y,a_z)\cdot(b_x,b_y,b_z)=a_xb_x + a_yb_y + a_zb_z$$ $$\bold a \cdot \bold b = |\bold a||\bold b| \cos \theta$$

如果$\bold a \cdot \bold b = 0$则 ab 互相正交

$$|\bold a \times \bold b| = |\bold a||\bold b|\sin \theta$$ $$\mathrm{Prj}_{\bold b}\bold a=\frac{\bold a \cdot \bold b}{|\bold b|}$$

点$P_0(x_0,y_0,z_0)$到平面$Ax+By+Cz+D=0$的距离是 $\frac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}$

球面$x^2 + y^2 + z^2 = a^2$

单叶双曲面（腰鼓形）$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$

双叶双曲面 $\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$

抛物面$x^2+y^2=z$

锥面$z=\sqrt{x^2+y^2}$

马鞍面$z=xy$

方向导数和梯度的关系$\frac{\partial u}{\partial \bold l} \vert_{p_0}=|\bold{grad} u\vert_{p_0}|\cos \theta$

散度 $\mathrm{div} \bold{A}=\frac{\partial{P}}{\partial{x}}+\frac{\partial{Q}}{\partial{y}}+\frac{\partial{R}}{\partial{z}}$

旋度 $\mathrm{rot} \bold{A}=\begin{vmatrix} \bold{i}& \bold{j} & \bold{k} \\ \frac{\partial}{\partial{x}} & \frac{\partial}{\partial{y}} & \frac{\partial}{\partial{z}} \\ P & Q & R \end{vmatrix}$