深度理解_微积分_偏导和空间向量

迈不过友情╰ 2022-10-22 12:52 192阅读 0赞

# 空间曲线，空间曲面 #

空间曲线的两种方式：①参数式②两面相交式

空间曲线一点的切线：①参数式：对x，y，z求t的偏导，**![\\frac\{x-x\_\{0\}\}\{x\{\}'(t)\}=\\frac\{y-y\_\{0\}\}\{y\{\}'(t)\}=\\frac\{z-z\_\{0\}\}\{z\{\}'(t)\}][frac_x-x_0_x_t_frac_y-y_0_y_t_frac_z-z_0_z_t]**②两面相交式：两个方程对x求导，解出两个偏导，切线：**![\\frac\{x-x\_\{0\}\}\{1\}=\\frac\{y-y\_\{0\}\}\{F\_\{y\}(M\_\{0\})\}=\\frac\{z-z\_\{0\}\}\{F\_\{z\}(M\_\{0\})\}][frac_x-x_0_1_frac_y-y_0_F_y_M_0_frac_z-z_0_F_z_M_0]**

空间曲面的表现方式：F(x,y,z)=0

二次曲面的一点的切线：![ax^2+by^2+cz^2=0][ax_2_by_2_cz_2_0]          ![axx\_\{0\}+byy\_\{0\}+czz\_\{0\}=0][axx_0_byy_0_czz_0_0]

空间曲面的切平面方程**：![F\_\{x\}(M\_\{0\})(x-x\_\{0\})+F\_\{y\}(M\_\{0\})(y-y\_\{0\})+F\_\{z\}(M\_\{0\})(z-z\_\{0\})=0][F_x_M_0_x-x_0_F_y_M_0_y-y_0_F_z_M_0_z-z_0_0]**

[frac_x-x_0_x_t_frac_y-y_0_y_t_frac_z-z_0_z_t]: https://private.codecogs.com/gif.latex?%5Cfrac%7Bx-x_%7B0%7D%7D%7Bx%7B%7D%27%28t%29%7D%3D%5Cfrac%7By-y_%7B0%7D%7D%7By%7B%7D%27%28t%29%7D%3D%5Cfrac%7Bz-z_%7B0%7D%7D%7Bz%7B%7D%27%28t%29%7D
[frac_x-x_0_1_frac_y-y_0_F_y_M_0_frac_z-z_0_F_z_M_0]: https://private.codecogs.com/gif.latex?%5Cfrac%7Bx-x_%7B0%7D%7D%7B1%7D%3D%5Cfrac%7By-y_%7B0%7D%7D%7BF_%7By%7D%28M_%7B0%7D%29%7D%3D%5Cfrac%7Bz-z_%7B0%7D%7D%7BF_%7Bz%7D%28M_%7B0%7D%29%7D
[ax_2_by_2_cz_2_0]: https://private.codecogs.com/gif.latex?ax%5E2&plus;by%5E2&plus;cz%5E2%3D0
[axx_0_byy_0_czz_0_0]: https://private.codecogs.com/gif.latex?axx_%7B0%7D&plus;byy_%7B0%7D&plus;czz_%7B0%7D%3D0
[F_x_M_0_x-x_0_F_y_M_0_y-y_0_F_z_M_0_z-z_0_0]: https://private.codecogs.com/gif.latex?F_%7Bx%7D%28M_%7B0%7D%29%28x-x_%7B0%7D%29&plus;F_%7By%7D%28M_%7B0%7D%29%28y-y_%7B0%7D%29&plus;F_%7Bz%7D%28M_%7B0%7D%29%28z-z_%7B0%7D%29%3D0