曲线的切线与法平面

曲线 Γ : { x = φ ( t ) y = ψ ( t ) z = ω ( t ) \Gamma:\begin{cases}x=\varphi(t) \\ y=\psi(t) \\ z=\omega(t) \end{cases} Γ:⎩⎪⎨⎪⎧​x=φ(t)y=ψ(t)z=ω(t)​， t ∈ [ α , β ] t\in[\alpha,\beta] t∈[α,β]

切线方程： x − x 0 φ ′ ( t 0 ) = y − y 0 ψ ′ ( t 0 ) = z − z 0 ω ′ ( t 0 ) \frac{x-x_{0}}{\varphi'(t_{0})}=\frac{y-y_{0}}{\psi'(t_{0})}=\frac{z-z_{0}}{\omega'(t_{0})} φ′(t0​)x−x0​​=ψ′(t0​)y−y0​​=ω′(t0​)z−z0​​

法平面方程： φ ′ ( t 0 ) ( x − x 0 ) + ψ ′ ( t 0 ) ( y − y 0 ) + ω ′ ( t 0 ) ( z − z 0 ) = 0 \varphi'(t_{0})(x-x_{0})+\psi'(t_{0})(y-y_{0})+\omega'(t_{0})(z-z_{0})=0 φ′(t0​)(x−x0​)+ψ′(t0​)(y−y0​)+ω′(t0​)(z−z0​)=0

曲面的切平面与法线

曲面 Σ : x = F ( x , y , z ) = 0 \Sigma:x=F(x,y,z)=0 Σ:x=F(x,y,z)=0

切平面方程： F x ( x 0 , y 0 , z 0 ) ( x − x 0 ) + F y ( x 0 , y 0 , z 0 ) ( y − y 0 ) + F z ( x 0 , y 0 , z 0 ) ( z − z 0 ) = 0 F_{x}(x_{0},y_{0},z_{0})(x-x_{0})+F_{y}(x_{0},y_{0},z_{0})(y-y_{0})+F_{z}(x_{0},y_{0},z_{0})(z-z_{0})=0 Fx​(x0​,y0​,z0​)(x−x0​)+Fy​(x0​,y0​,z0​)(y−y0​)+Fz​(x0​,y0​,z0​)(z−z0​)=0

法线方程： x − x 0 F x ( x 0 , y 0 , z 0 ) = y − y 0 F y ( x 0 , y 0 , z 0 ) = z − z 0 F z ( x 0 , y 0 , z 0 ) \frac{x-x_{0}}{F_{x}(x_{0},y_{0},z_{0})}=\frac{y-y_{0}}{F_{y}(x_{0},y_{0},z_{0})}=\frac{z-z_{0}}{F_{z}(x_{0},y_{0},z_{0})} Fx​(x0​,y0​,z0​)x−x0​​=Fy​(x0​,y0​,z0​)y−y0​​=Fz​(x0​,y0​,z0​)z−z0​​

标签：frac,法平面,切线,t0,y0,x0,高等数学,z0
来源： https://blog.csdn.net/weixin_43896318/article/details/114842328