多元函数积分学

向量的运算

(ax,ay,az)⋅(bx,by,bz)=axbx+ayby+azbz(a_x,a_y,a_z)\cdot(b_x,b_y,b_z)=a_xb_x + a_yb_y + a_zb_z a⋅b=∣a∣∣b∣cos⁡θ\bold a \cdot \bold b = |\bold a||\bold b| \cos \theta

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

∣a×b∣=∣a∣∣b∣sin⁡θ|\bold a \times \bold b| = |\bold a||\bold b|\sin \theta Prjba=a⋅b∣b∣\mathrm{Prj}_{\bold b}\bold a=\frac{\bold a \cdot \bold b}{|\bold b|}

点P0(x0,y0,z0)P_0(x_0,y_0,z_0)到平面Ax+By+Cz+D=0Ax+By+Cz+D=0的距离是 ∣Ax0+By0+Cz0+D∣A2+B2+C2\frac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}

球面x2+y2+z2=a2x^2 + y^2 + z^2 = a^2

单叶双曲面（腰鼓形）x2a2+y2b2−z2c2=1\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1

双叶双曲面 x2a2−y2b2−z2c2=1\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1

抛物面x2+y2=zx^2+y^2=z

锥面z=x2+y2z=\sqrt{x^2+y^2}

马鞍面z=xyz=xy

方向导数和梯度的关系∂u∂l∣p0=∣gradu∣p0∣cos⁡θ\frac{\partial u}{\partial \bold l} \vert_{p_0}=|\bold{grad} u\vert_{p_0}|\cos \theta

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

旋度 rotA=∣ijk∂∂x∂∂y∂∂zPQR∣\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}