微积分学习笔记(十一)多元函数积分

空间曲线的切向量

\left\{\begin{array}{l}x=x(t) \\ y=y(t) \\ z=z(t)\end{array}\right.

切向量就是割向量取极限:

(\lim_{\Delta t \rightarrow 0}\frac{x\left(t_0+\Delta t\right)-x\left(t_0\right)}{\Delta t}, \lim_{\Delta t \rightarrow 0}\frac{y\left(t_0+\Delta t\right)-y\left(t_0\right)}{\Delta t},\lim_{\Delta t \rightarrow 0}\frac{z\left(t_0+\Delta t\right)-z\left(t_0\right)}{\Delta t})

切向量为:

(x'(t_0) , y'(t_0),z'(t_0))

空间曲面的法向量

F(x,y,z)=0

平面内过M点 \left(x\left(t_0\right), y\left(t_0\right), z\left(t_0\right)\right) 曲线为:

\left\{\begin{array}{l}x=x(t) \\ y=y(t) \\ z=z(t)\end{array}\right.

代入到平面:

F(x\left(t_0\right), y\left(t_0\right), z\left(t_0\right))=0

求导得:

F_xx'(t_0) +F_y y'(t_0)+F_zz'(t_0)=0

整理成:

(F_x, F_y,F_z)\cdot (x'(t_0) , y'(t_0),z'(t_0))=0

向量 (F_x, F_y,F_z) 与任意的平面内曲线都垂直,所以法向量为:

(F_x, F_y,F_z)

空间曲面面积

dS=\cos \gamma dxdy

\cos \gamma 是xy平面与曲面的夹角。 \vec{n} \vec{z} 分别是曲线法向量和xy平面的法向量,则夹角为:

\cos\gamma=\frac{\vec{n} \cdot \vec{z}}{|\vec{n}| \cdot|\vec{z}|}

posted @ 2025/07/23 02:18:25