空间曲线的切线与法平面
参数方程 ( x ( t ) , y ( t ) , z ( t ) ) (x(t),y(t),z(t)) (x(t),y(t),z(t))
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切方向为 ( x ′ ( t ) , y ′ ( t ) , z ′ ( t ) ) (x'(t),y'(t),z'(t)) (x′(t),y′(t),z′(t))，
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在 ( x 0 , y 0 , z 0 ) = ( x ( t 0 ) , y ( t 0 ) , z ( t 0 ) ) (x_0,y_0,z_0)=(x(t_0),y(t_0),z(t_0)) (x0,y0,z0)=(x(t0),y(t0),z(t0)) 处的切线方程为
x − x 0 x ′ ( t 0 ) = y − y 0 y ′ ( t 0 ) = z − z 0 z ′ ( t 0 ) \frac{x-x_0}{x'(t_0)}=\frac{y-y_0}{y'(t_0)}=\frac{z-z_0}{z'(t_0)} x′(t0)x−x0=y′(t0)y−y0=z′(t0)z−z0
- 法线方程
x ′ ( t 0 ) ( x − x 0 ) + y ′ ( t 0 ) ( y − y 0 ) + z ′ ( t 0 ) ( z − z 0 ) = 0 {x'(t_0)}(x-x_0)+y'(t_0)(y-y_0)+z'(t_0) (z-z_0)=0 x′(t0)(x−x0)+y′(t0)(y−y0)+z′(t0)(z−z0)=0
参数方程特例 ( x , y ( x ) , z ( x ) ) (x,y(x),z(x)) (x,y(x),z(x))
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切方向为 ( 1 , y ′ ( x ) , z ′ ( x ) ) (1,y'(x),z'(x)) (1,y′(x),z′(x))，
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在 ( x 0 , y 0 , z 0 ) = ( x 0 , y ( x 0 ) , z ( x 0 ) ) (x_0,y_0,z_0)=(x_0,y(x_0),z(x_0)) (x0,y0,z0)=(x0,y(x0),z(x0)) 处的切线方程为
x − x 0 = y − y 0 y ′ ( x 0 ) = z − z 0 z ′ ( x 0 ) {x-x_0}=\frac{y-y_0}{y'(x_0)}=\frac{z-z_0}{z'(x_0)} x−x0=y′(x0)y−y0=z′(x0)z−z0
- 法平面方程
( x − x 0 ) + y ′ ( x 0 ) ( y − y 0 ) + z ′ ( x 0 ) ( z − z 0 ) = 0 (x-x_0)+y'(x_0)(y-y_0)+z'(x_0) (z-z_0)=0 (x−x0)+y′(x0)(y−y0)+z′(x0)(z−z0)=0
一般方程 F ( x , y , z ) = 0 , G ( x , y , z ) = 0 F(x,y,z)=0, G(x,y,z)=0 F(x,y,z)=0,G(x,y,z)=0
由一般方程可以确立隐函数 ( y = y ( x ) , z = z ( x ) ) (y=y(x),z=z(x)) (y=y(x),z=z(x)),
0 = ∂ F ( x , y , z ) ∂ x + ∂ F ( x , y , z ) ∂ y d y d x + ∂ F ( x , y , z ) ∂ z d z d x 0=\frac{\partial F(x,y,z)}{\partial x} + \frac{\partial F(x,y,z)}{\partial y}\frac{dy}{dx}+ \frac{\partial F(x,y,z)}{\partial z}\frac{dz}{dx} 0=∂x∂F(x,y,z)+∂y∂F(x,y,z)dxdy+∂z∂F(x,y,z)dxdz
0 = ∂ G ( x , y , z ) ∂ x + ∂ G ( x , y , z ) ∂ y d y d x + ∂ G ( x , y , z ) ∂ z d z d x 0=\frac{\partial G(x,y,z)}{\partial x} + \frac{\partial G(x,y,z)}{\partial y}\frac{dy}{dx}+ \frac{\partial G(x,y,z)}{\partial z}\frac{dz}{dx} 0=∂x∂G(x,y,z)+∂y∂G(x,y,z)dxdy+∂z∂G(x,y,z)dxdz
因此
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\frac{dy}{dx}=-\frac{1}{J} \frac{\partial(F,G)}{\partial (x,z)}
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\frac{dz}{dx}=-\frac{1}{J} \frac{\partial(F,G)}{\partial (y,x)}
dxdz=−J1∂(y,x)∂(F,G)
回代到参数方程的特例中
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在 P 0 = ( x 0 , y 0 , z 0 ) P_0=(x_0,y_0,z_0) P0=(x0,y0,z0) 的法向量切线方程
x − x 0 = y − y 0 y ′ ∣ P 0 = z − z 0 z ′ ∣ P 0 {x-x_0}=\frac{y-y_0}{y'|_{P_0}}=\frac{z-z_0}{z'|_{P_0}} x−x0=y′∣P0y−y0=z′∣P0z−z0
即
x − x 0 ∂ ( F , G ) ∂ ( y , z ) ∣ P 0 = y − y 0 ∂ ( F , G ) ∂ ( z , x ) ∣ P 0 = z − z 0 ∂ ( F , G ) ∂ ( x , y ) ∣ P 0 \frac{x-x_0}{\frac{\partial (F,G)}{\partial (y,z)}|_{P_0}}=\frac{y-y_0}{\frac{\partial(F,G)}{\partial (z,x)}|_{P_0}}=\frac{z-z_0}{\frac{\partial(F,G)}{\partial (x,y)}|_{P_0}} ∂(y,z)∂(F,G)∣P0x−x0=∂(z,x)∂(F,G)∣P0y−y0=∂(x,y)∂(F,G)∣P0z−z0 -
在 P 0 = ( x 0 , y 0 , z 0 ) P_0=(x_0,y_0,z_0) P0=(x0,y0,z0) 的法向量法平面方程
∂ ( F , G ) ∂ ( y , z ) ∣ P 0 ( x − x 0 ) + ∂ ( F , G ) ∂ ( z , x ) ∣ P 0 ( y − y 0 ) + ∂ ( F , G ) ∂ ( x , y ) ∣ P 0 ( z − z 0 ) = 0 {\frac{\partial (F,G)}{\partial (y,z)}|_{P_0}}(x-x_0)+{\frac{\partial(F,G)}{\partial (z,x)}|_{P_0}}(y-y_0)+{\frac{\partial(F,G)}{\partial (x,y)}|_{P_0}}(z-z_0)=0 ∂(y,z)∂(F,G)∣P0(x−x0)+∂(z,x)∂(F,G)∣P0(y−y0)+∂(x,y)∂(F,G)∣P0(z−z0)=0
即
∣ x − x 0 y − y 0 z − z 0 ∂ F ∂ x ∣ P 0 ∂ F ∂ y ∣ P 0 ∂ F ∂ z ∣ P 0 ∂ G ∂ x ∣ P 0 ∂ G ∂ y ∣ P 0 ∂ G ∂ z ∣ P 0 ∣ = 0 \left| \begin{matrix} x-x_0 & y-y_0 & z-z_0\\ \frac{\partial F}{\partial x}|_{P_0} &\frac{\partial F}{\partial y}|_{P_0} & \frac{\partial F}{\partial z}|_{P_0}\\ \frac{\partial G}{\partial x}|_{P_0} &\frac{\partial G}{\partial y}|_{P_0} & \frac{\partial G}{\partial z}|_{P_0}\\ \end{matrix}\right|=0 x−x0∂x∂F∣P0∂x∂G∣P0y−y0∂y∂F∣P0∂y∂G∣P0z−z0∂z∂F∣P0∂z∂G∣P0 =0
空间曲面的切平面与法线
一般式 F ( x , y , z ) = 0 F(x,y,z)=0 F(x,y,z)=0
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在 P 0 = ( x 0 , y 0 , z 0 ) P_0=(x_0,y_0,z_0) P0=(x0,y0,z0) 的法向量 n = ( ∂ F ∂ x , ∂ F ∂ y , ∂ F ∂ z ) ∣ P 0 n=(\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z})|_{P_0} n=(∂x∂F,∂y∂F,∂z∂F)∣P0
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在 P 0 = ( x 0 , y 0 , z 0 ) P_0=(x_0,y_0,z_0) P0=(x0,y0,z0) 的切平面 ∂ F ∂ x ∣ P 0 ⋅ ( x − x 0 ) + ∂ F ∂ y ∣ P 0 ⋅ ( y − y 0 ) + ∂ F ∂ z ∣ P 0 ⋅ ( z − z 0 ) = 0 \frac{\partial F}{\partial x}|_{P_0} \cdot (x-x_0)+\frac{\partial F}{\partial y}|_{P_0}\cdot(y-y_0)+\frac{\partial F}{\partial z}|_{P_0}\cdot(z-z_0)=0 ∂x∂F∣P0⋅(x−x0)+∂y∂F∣P0⋅(y−y0)+∂z∂F∣P0⋅(z−z0)=0
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在 P 0 = ( x 0 , y 0 , z 0 ) P_0=(x_0,y_0,z_0) P0=(x0,y0,z0) 的法线 x − x 0 ∂ F ∂ x ∣ P 0 + y − y 0 ∂ F ∂ y ∣ P 0 + z − z 0 ∂ F ∂ z ∣ P 0 = 0 \frac{x-x_0}{\frac{\partial F}{\partial x}|_{P_0}}+\frac{y-y_0}{\frac{\partial F}{\partial y}|_{P_0}} + \frac{z-z_0}{\frac{\partial F}{\partial z}|_{P_0}}=0 ∂x∂F∣P0x−x0+∂y∂F∣P0y−y0+∂z∂F∣P0z−z0=0
二元函数 z = f ( x , y ) z=f(x,y) z=f(x,y)
设 F = f ( x , y ) − z F=f(x,y)-z F=f(x,y)−z
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在 P 0 = ( x 0 , y 0 , f ( x 0 , y 0 ) ) P_0=(x_0,y_0,f(x_0,y_0)) P0=(x0,y0,f(x0,y0)) 的法向量 n = ( ∂ f ∂ x , ∂ f ∂ y , − 1 ) ∣ P 0 n=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},-1)|_{P_0} n=(∂x∂f,∂y∂f,−1)∣P0
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在 P 0 P_0 P0 出的切平面
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\frac{\partial f}{\partial x}|_{P_0} \cdot (x-x_0)+\frac{\partial f}{\partial y}|_{P_0}\cdot(y-y_0)-(z-z_0)=0
∂x∂f∣P0⋅(x−x0)+∂y∂f∣P0⋅(y−y0)−(z−z0)=0
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z=f(x_0,y_0)+\frac{\partial f}{\partial x}|_{P_0} \cdot (x-x_0)+\frac{\partial f}{\partial y}|_{P_0}\cdot(y-y_0)
z=f(x0,y0)+∂x∂f∣P0⋅(x−x0)+∂y∂f∣P0⋅(y−y0)
- 在
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− ( z − z 0 ) = x − x 0 ∂ f ∂ x ∣ P 0 = y − y 0 ∂ f ∂ y ∣ P 0 -(z-z_0)=\frac{x-x_0}{\frac{\partial f}{\partial x}|_{P_0}}=\frac{y-y_0}{\frac{\partial f}{\partial y}|_{P_0}} −(z−z0)=∂x∂f∣P0x−x0=∂y∂f∣P0y−y0
(选看) n n n 元函数 z = f ( x ) z=f(x) z=f(x), x = ( x 1 , ⋯ , x n ) x=(x_1,\cdots,x_n) x=(x1,⋯,xn), x ˉ = ( x ˉ 1 , ⋯ , x ˉ n ) \bar{x}=(\bar{x}_1,\cdots,\bar{x}_n) xˉ=(xˉ1,⋯,xˉn)
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切平面: z = f ( x ˉ ) + ⟨ ∇ f ( x ˉ ) , x − x ˉ ⟩ z=f(\bar{x})+\langle \nabla f(\bar{x}), x-\bar{x}\rangle z=f(xˉ)+⟨∇f(xˉ),x−xˉ⟩
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法平面: − ( z − f ( x ˉ ) ) = x 1 − x ˉ 1 ∂ f ∂ x 1 ∣ x ˉ = ⋯ = x n − x ˉ n ∂ f ∂ x n ∣ x ˉ -(z-f(\bar{x}))=\frac{x_1-\bar{x}_1}{\frac{\partial f}{\partial x_1}|_{\bar{x}}}=\cdots=\frac{x_n-\bar{x}_n}{\frac{\partial f}{\partial x_n}|_{\bar{x}}} −(z−f(xˉ))=∂x1∂f∣xˉx1−xˉ1=⋯=∂xn∂f∣xˉxn−xˉn
参数方程 ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) (x(u,v),y(u,v),z(u,v)) (x(u,v),y(u,v),z(u,v))
可以确定隐函数 u ( x , y ) u(x,y) u(x,y) 与 v ( x , y ) v(x,y) v(x,y),
J = ∂ ( x , y ) ∂ ( u , v ) J=\frac{\partial (x,y)}{\partial (u,v)} J=∂(u,v)∂(x,y),
因此 z = z ( u ( x , y ) , v ( x , y ) ) z=z(u(x,y),v(x,y)) z=z(u(x,y),v(x,y))，由反函数的偏微分
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\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\frac{\partial v}{\partial x}= \frac{1}{J}\frac{\partial (z,y)}{\partial (u,v)}
∂x∂z=∂u∂z∂x∂u+∂v∂z∂x∂v=J1∂(u,v)∂(z,y)
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\frac{\partial z}{\partial y}=\frac{\partial z}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial z}{\partial v}\frac{\partial v}{\partial y}=\frac{1}{J}\frac{\partial (x,z)}{\partial (u,v)}
∂y∂z=∂u∂z∂y∂u+∂v∂z∂y∂v=J1∂(u,v)∂(x,z)
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∂z∂z=1
带入法向量中
n = ( ∂ ( y , z ) ∂ ( u , v ) , ∂ ( z , x ) ∂ ( u , v ) , ∂ ( x , y ) ∂ ( u , v ) ) n=(\frac{\partial(y,z)}{\partial (u,v)},\frac{\partial(z,x)}{\partial (u,v)}, \frac{\partial (x,y)}{\partial (u,v)}) n=(∂(u,v)∂(y,z),∂(u,v)∂(z,x),∂(u,v)∂(x,y))
切平面方程
⟨ n , P 0 P ⟩ = 0 \langle n, P_0 P\rangle =0 ⟨n,P0P⟩=0
∣ x − x 0 y − y 0 z − z 0 ∂ x ∂ u ∣ P 0 ∂ y ∂ u ∣ P 0 ∂ z ∂ u ∣ P 0 ∂ x ∂ v ∣ P 0 ∂ y ∂ v ∣ P 0 ∂ z ∂ v ∣ P 0 ∣ = 0 \left|\begin{matrix} x-x_0 & y-y_0 & z-z_0\\ \frac{\partial x}{\partial u}|_{P_0} &\frac{\partial y}{\partial u}|_{P_0}&\frac{\partial z}{\partial u}|_{P_0}\\ \frac{\partial x}{\partial v}|_{P_0} &\frac{\partial y}{\partial v}|_{P_0}&\frac{\partial z}{\partial v}|_{P_0} \end{matrix}\right| =0 x−x0∂u∂x∣P0∂v∂x∣P0y−y0∂u∂y∣P0∂v∂y∣P0z−z0∂u∂z∣P0∂v∂z∣P0 =0
法线方程
x − x 0 ∂ ( y , z ) ∂ ( u , v ) = y − y 0 ∂ ( z , x ) ∂ ( u , v ) = z − z 0 ∂ ( x , y ) ∂ ( u , v ) \frac{x-x_0}{\frac{\partial(y,z)}{\partial (u,v)}} =\frac{y-y_0}{\frac{\partial(z,x)}{\partial (u,v)}}=\frac{z-z_0}{\frac{\partial (x,y)}{\partial (u,v)}} ∂(u,v)∂(y,z)x−x0=∂(u,v)∂(z,x)y−y0=∂(u,v)∂(x,y)z−z0
方向导数
给定向量的方向角 ( α , β , γ ) (\alpha,\beta,\gamma) (α,β,γ), l = ( cos ( α ) , cos ( β ) , cos ( γ ) ) l=(\cos(\alpha), \cos(\beta),\cos(\gamma)) l=(cos(α),cos(β),cos(γ)), f f f在 ( x 0 , y 0 , z 0 ) (x_0,y_0,z_0) (x0,y0,z0) 沿着方向 l l l 的方向导数为
∂ f ∂ l = lim t → 0 f ( x 0 + t cos ( α ) , y 0 + t cos ( β ) , z 0 + t cos ( γ ) ) − f ( x 0 , y 0 , z 0 ) t \frac{\partial f}{\partial l}=\lim_{t\to 0}\frac{f(x_0+t\cos(\alpha), y_0+t\cos(\beta),z_0+t\cos(\gamma))-f(x_0,y_0,z_0)}{t} ∂l∂f=limt→0tf(x0+tcos(α),y0+tcos(β),z0+tcos(γ))−f(x0,y0,z0)
- 若函数 f f f 可微 ∂ f ∂ l = ⟨ ∇ f ( x ) , l ⟩ = ∂ f ∂ x cos ( α ) + ∂ f ∂ y cos ( β ) + ∂ f ∂ z cos ( γ ) \frac{\partial f}{\partial l}= \langle \nabla f(x), l\rangle= \frac{\partial f}{\partial x}\cos(\alpha)+\frac{\partial f}{\partial y}\cos(\beta)+\frac{\partial f}{\partial z}\cos(\gamma) ∂l∂f=⟨∇f(x),l⟩=∂x∂fcos(α)+∂y∂fcos(β)+∂z∂fcos(γ)
(选看) 其他方向导数
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Gatueax 方向导数： 在 x x x 处沿着方向 d d d 的 Gatueax 方向导数， 如果极限存在 f ′ ( x ; d ) = lim t → 0 f ( x + t d ) − f ( x ) t f'(x;d)=\lim_{t\to 0} \frac{f(x+td)-f(x)}{t} f′(x;d)=limt→0tf(x+td)−f(x)
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Hadmard 导数： 在 x x x 处沿着方向 d d d 的 Hadmard 方向导数， 如果极限存在 f o ( x ; d ) = lim t → 0 , h → d f ( x + t h ) − f ( x ) t f^o(x;d)=\lim_{t\to 0, h\to d} \frac{f(x+t h)-f(x)}{t} fo(x;d)=limt→0,h→dtf(x+th)−f(x)
多元函数求极值
极小值定义
在 P 0 P_0 P0 的某个邻域 U ( P 0 ) U(P_0) U(P0)内, f ( P ) ≥ f ( P 0 ) f(P)\geq f(P_0) f(P)≥f(P0), ∀ P ∈ U ( P 0 ) \forall P\in U(P_0) ∀P∈U(P0)
极大值定义
在 P 0 P_0 P0 的某个邻域 U ( P 0 ) U(P_0) U(P0)内, f ( P ) ≤ f ( P 0 ) f(P)\leq f(P_0) f(P)≤f(P0), ∀ P ∈ U ( P 0 ) \forall P\in U(P_0) ∀P∈U(P0)
二元函数
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一阶必要性: 函数 f f f 可微， 如果在 ( x 0 , y 0 ) (x_0,y_0) (x0,y0) 处取得极值， 则 ∂ f ( x 0 , y 0 ) ∂ x = ∂ f ( x 0 , y 0 ) ∂ y = 0 \frac{\partial f(x_0,y_0)}{\partial x}=\frac{\partial f(x_0,y_0)}{\partial y}=0 ∂x∂f(x0,y0)=∂y∂f(x0,y0)=0.
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二阶必要性: 函数 f f f 可微， 如果在 ( x 0 , y 0 ) (x_0,y_0) (x0,y0) 处取得极小值， 则 ∂ f ( x 0 , y 0 ) ∂ x = ∂ f ( x 0 , y 0 ) ∂ y = 0 \frac{\partial f(x_0,y_0)}{\partial x}=\frac{\partial f(x_0,y_0)}{\partial y}=0 ∂x∂f(x0,y0)=∂y∂f(x0,y0)=0， ∂ 2 f ∂ x 2 ≥ 0 \frac{\partial^2 f}{\partial x^2}\geq 0 ∂x2∂2f≥0, ∣ ∂ 2 f ∂ x 2 ∂ 2 f ∂ x ∂ y ∂ 2 f ∂ x ∂ y ∂ 2 f ∂ y 2 ∣ ≥ 0 \left|\begin{matrix}\frac{\partial^2 f}{\partial x^2} &\frac{\partial^2 f}{\partial x\partial y}\\ \frac{\partial^2 f}{\partial x\partial y} &\frac{\partial^2 f}{\partial y^2} \end{matrix}\right|\geq 0 ∂x2∂2f∂x∂y∂2f∂x∂y∂2f∂y2∂2f ≥0.
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二阶必要性: 函数 f f f 可微， 如果在 ( x 0 , y 0 ) (x_0,y_0) (x0,y0) 处取得极大值， 则 ∂ f ( x 0 , y 0 ) ∂ x = ∂ f ( x 0 , y 0 ) ∂ y = 0 \frac{\partial f(x_0,y_0)}{\partial x}=\frac{\partial f(x_0,y_0)}{\partial y}=0 ∂x∂f(x0,y0)=∂y∂f(x0,y0)=0， ∂ 2 f ∂ x 2 ≤ 0 \frac{\partial^2 f}{\partial x^2}\leq 0 ∂x2∂2f≤0, ∣ ∂ 2 f ∂ x 2 ∂ 2 f ∂ x ∂ y ∂ 2 f ∂ x ∂ y ∂ 2 f ∂ y 2 ∣ ≥ 0 \left|\begin{matrix}\frac{\partial^2 f}{\partial x^2} &\frac{\partial^2 f}{\partial x\partial y}\\ \frac{\partial^2 f}{\partial x\partial y} &\frac{\partial^2 f}{\partial y^2} \end{matrix}\right|\geq 0 ∂x2∂2f∂x∂y∂2f∂x∂y∂2f∂y2∂2f ≥0.
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二阶充分性: 函数 f f f 可微， 如果 ∂ f ( x 0 , y 0 ) ∂ x = ∂ f ( x 0 , y 0 ) ∂ y = 0 \frac{\partial f(x_0,y_0)}{\partial x}=\frac{\partial f(x_0,y_0)}{\partial y}=0 ∂x∂f(x0,y0)=∂y∂f(x0,y0)=0， ∂ 2 f ∂ x 2 > 0 \frac{\partial^2 f}{\partial x^2}> 0 ∂x2∂2f>0, ∣ ∂ 2 f ∂ x 2 ∂ 2 f ∂ x ∂ y ∂ 2 f ∂ x ∂ y ∂ 2 f ∂ y 2 ∣ > 0 \left|\begin{matrix}\frac{\partial^2 f}{\partial x^2} &\frac{\partial^2 f}{\partial x\partial y}\\ \frac{\partial^2 f}{\partial x\partial y} &\frac{\partial^2 f}{\partial y^2} \end{matrix}\right|> 0 ∂x2∂2f∂x∂y∂2f∂x∂y∂2f∂y2∂2f >0, 则在 ( x 0 , y 0 ) (x_0,y_0) (x0,y0) 处取得极小值.
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二阶充分性: 函数 f f f 可微， 如果 ∂ f ( x 0 , y 0 ) ∂ x = ∂ f ( x 0 , y 0 ) ∂ y = 0 \frac{\partial f(x_0,y_0)}{\partial x}=\frac{\partial f(x_0,y_0)}{\partial y}=0 ∂x∂f(x0,y0)=∂y∂f(x0,y0)=0， ∂ 2 f ∂ x 2 < 0 \frac{\partial^2 f}{\partial x^2}< 0 ∂x2∂2f<0, ∣ ∂ 2 f ∂ x 2 ∂ 2 f ∂ x ∂ y ∂ 2 f ∂ x ∂ y ∂ 2 f ∂ y 2 ∣ > 0 \left|\begin{matrix}\frac{\partial^2 f}{\partial x^2} &\frac{\partial^2 f}{\partial x\partial y}\\ \frac{\partial^2 f}{\partial x\partial y} &\frac{\partial^2 f}{\partial y^2} \end{matrix}\right|> 0 ∂x2∂2f∂x∂y∂2f∂x∂y∂2f∂y2∂2f >0. 则在 ( x 0 , y 0 ) (x_0,y_0) (x0,y0) 处取得极大值
(选看) n 元函数
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一阶必要性: 函数 f f f 可微， 如果在 P 0 P_0 P0 处取得极值， 则 ∇ f ( P 0 ) = 0 \nabla f(P_0)=0 ∇f(P0)=0.
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二阶必要性: 函数 f f f 可微， 如果在 P 0 P_0 P0 处取得极小值， 则 ∇ f ( P 0 ) = 0 \nabla f(P_0)=0 ∇f(P0)=0， ∇ 2 f ( P 0 ) ⪰ 0 \nabla^2 f(P_0)\succeq 0 ∇2f(P0)⪰0 半正定的海瑟矩阵.
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二阶必要性: 函数 f f f 可微， 如果在 P 0 P_0 P0 处取得极大值， 则 ∇ f ( P 0 ) = 0 \nabla f(P_0)=0 ∇f(P0)=0， ∇ 2 f ( P 0 ) ⪯ 0 \nabla^2 f(P_0)\preceq 0 ∇2f(P0)⪯0 半负定的海瑟矩阵.
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二阶充分性: 函数 f f f 可微， 如果 ∇ f ( P 0 ) = 0 \nabla f(P_0)=0 ∇f(P0)=0， ∇ 2 f ( P 0 ) ≻ 0 \nabla^2 f(P_0)\succ 0 ∇2f(P0)≻0 正定海瑟矩阵， 则 P 0 P_0 P0 处取得极小值.
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二阶充分性: 函数 f f f 可微， 如果 ∇ f ( P 0 ) = 0 \nabla f(P_0)=0 ∇f(P0)=0， ∇ 2 f ( P 0 ) ≺ 0 \nabla^2 f(P_0)\prec 0 ∇2f(P0)≺0 负定海瑟矩阵， 则 P 0 P_0 P0 处取得极大值.
条件极值
min
f
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\min f(x,y,z)
minf(x,y,z)
s
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t
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ϕ
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y
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z
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=
0
\mathrm{s.t.} ~~\phi(x,y,z)=0
s.t. ϕ(x,y,z)=0
ψ
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x
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y
,
z
)
=
0
\quad\quad \psi(x,y,z)=0
ψ(x,y,z)=0
设 拉格朗日函数 L ( x , y , z ; λ , μ ) = f ( x , y , z ) + λ ϕ ( x , y , z ) + μ ψ ( x , y , z ) L(x,y,z;\lambda,\mu)= f(x,y,z)+\lambda \phi(x,y,z)+\mu \psi(x,y,z) L(x,y,z;λ,μ)=f(x,y,z)+λϕ(x,y,z)+μψ(x,y,z)
联立梯度与等式约束得到方程组
0
=
∂
∂
x
f
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y
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+
λ
∂
∂
x
ϕ
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+
μ
∂
∂
x
ψ
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x
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0=\frac{\partial}{\partial x}f(x,y,z)+\lambda \frac{\partial}{\partial x}\phi(x,y,z)+\mu \frac{\partial}{\partial x}\psi(x,y,z)
0=∂x∂f(x,y,z)+λ∂x∂ϕ(x,y,z)+μ∂x∂ψ(x,y,z)
0
=
∂
∂
y
f
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+
λ
∂
∂
y
ϕ
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+
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∂
∂
y
ψ
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0=\frac{\partial}{\partial y}f(x,y,z)+\lambda \frac{\partial}{\partial y}\phi(x,y,z)+\mu \frac{\partial}{\partial y}\psi(x,y,z)
0=∂y∂f(x,y,z)+λ∂y∂ϕ(x,y,z)+μ∂y∂ψ(x,y,z)
0
=
∂
∂
z
f
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+
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∂
∂
z
ϕ
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+
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∂
∂
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ψ
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)
0=\frac{\partial}{\partial z}f(x,y,z)+\lambda \frac{\partial}{\partial z}\phi(x,y,z)+\mu \frac{\partial}{\partial z}\psi(x,y,z)
0=∂z∂f(x,y,z)+λ∂z∂ϕ(x,y,z)+μ∂z∂ψ(x,y,z)
0
=
ϕ
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x
,
y
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0=\phi(x,y,z)
0=ϕ(x,y,z)
0
=
ψ
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0=\psi(x,y,z)
0=ψ(x,y,z)
进一步判断 L L L 的二阶偏导判断其极大与极小性质.
n元函数条件极值
min
f
(
x
)
\min f(x)
minf(x)
s
.
t
.
g
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x
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=
0
\mathrm{s.t.}~~ g(x)=0
s.t. g(x)=0
L
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=
f
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+
g
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⊤
λ
L(x,\lambda)=f(x)+g(x)^\top\lambda
L(x,λ)=f(x)+g(x)⊤λ
一阶必要性条件
0
=
∇
x
L
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λ
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=
∇
x
f
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J
x
g
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∗
λ
0=\nabla_x L(x,\lambda)=\nabla_x f(x)+J_xg(x)^{*}\lambda
0=∇xL(x,λ)=∇xf(x)+Jxg(x)∗λ.
且
g
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x
)
=
0
g(x)=0
g(x)=0