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工科数学分析-3.26
2026-03-26
工数分
数学

设 A1B1=2a,b1G=x=QO,QP=DG=yA_1B_1=2a,b_1G=x=QO,QP=DG=y
,则 OP=a,x2+a2=y2,DA1=3aOP=a,x^2+a^2=y^2,DA_1=\sqrt{3}a
SA1B1C1D1E1F1=63a2,SA1B1G=ax,SAqGC1Q=23ayS_{A_1B_1C_1D_1E_1F_1}=6\sqrt{3}a^2,S_{A_1B_1G}=ax,S_{A_qGC_1Q}=2\sqrt{3}ay
问题归纳为，求，面积减少的最大值

{z=63a2+6ax−63ayx2+a2=y2,x≥0\begin{cases} z&=6\sqrt{3}a^2+6ax-6\sqrt{3}ay\\ x^2+a^2&=y^2,x\ge0 \end{cases}

L(x,y,λ)=63a2+6ax−63ay+λ(x2+a2−y2)L(x,y,\lambda)=6\sqrt{3}a^2+6ax-6\sqrt{3}ay+\lambda(x^2+a^2-y^2)
∇L=0\nabla L=0

{Lx=6a+2λ=0Ly=−63a−2λy=0Lλ=a2+a2−y2=0\begin{cases} L_x&=6a+2\lambda &=0\\ L_y&=-6\sqrt{3}a-2\lambda y&=0\\ L_{\lambda}&=a^2+a^2-y^2&=0 \end{cases}

解出
x=a2,y=3a2x=\frac{a}{\sqrt{2}},y=\frac{\sqrt{3}a}{\sqrt{2}}
z=6(3−2)a2z=6(\sqrt{3}-\sqrt{2})a^{2}
x→0+,y→a,z→0x\rightarrow0^+,y\rightarrow a,z\rightarrow 0
x→+∞,z=63a2+6a(x−3y)<63a2+6a(1−3)x→−∞x\rightarrow +\infty,z=6\sqrt{3}a^2+6a(x-\sqrt{3}y)<6\sqrt{3}a^2+6a(1-\sqrt{3})x\rightarrow -\infty
因此zz在约束内部点(a2,3a2)(\frac{a}{\sqrt{2}},\frac{\sqrt{3}a}{\sqrt{2}})取到最大值6(3−2)a26(\sqrt{3}-\sqrt{2})a^2
此时
tan⁡(∠A1GD)=PA1PG=3ay=2\tan(\angle A_1GD)=\frac{PA_1}{PG}=\frac{\sqrt{3}a}{y}=\sqrt{2}
tan⁡(∠A1GC1)=tan⁡(2∠A1GP)=−22\tan(\angle A_1GC_1)=\tan(2\angle A_1G_P)=-2\sqrt{2}
∠A1GC1≈109∘28′\angle A_1GC_1\approx 109^{\circ}28'

5.5 多元向量值函数的导数与微分#

nn元向量值函数F:A⊂Rn→RmF:A\subset R^n\rightarrow R^m
f(X)=(f1(X),f2(X),...,fm(X)),X∈Af(X)=(f_1(X),f_2(X),...,f_m(X)),X\in A
设a=(a1,a2,...,am)∈Rma=(a_1,a_2,...,a_m)\in R^m定义极限
lim⁡X→X0f(X)=a⇔lim⁡X→X0fi(X)=ai,∀i=1,2,..,m\lim_{X\rightarrow X_0}f(X)=a\Leftrightarrow \lim_{X\to X_0}f_i(X)=a_i,\forall i=1,2,..,m
若lim⁡X→X0f(X)=f(X0)\lim_{X\to X_0}f(X)=f(X_0),则称向量值函数在ff在X=X0X=X_0处连续

5.5.1 一元向量值函数#

对f(X)=(f1(X),...,fm(X))f(X)=(f_1(X),...,f_m(X)),定义导数
f′(X0)=(f1′(X0),...,fm′(X0))f'(X_0)=(f_{1}'(X_{0}),...,f_m'(X_0))
定义二阶导数
f′′(X0)=(f1′′(X0),...,fm′′(X0))f''(X_0)=(f_1''(X_0),...,f_m''(X_0))
若存在向量a=(a1,a2,...,am)a=(a_1,a_2,...,a_m)，使
f(X0+ΔX)=f(X0)+aΔX+o(ΔX)f(X_0+\Delta X)=f(X_0)+a\Delta X+o(\Delta X)
,则称ff在X=X0X=X_0处可微
ff在X0X_0处的微分df(X0)=aΔXdf(X_0)=a\Delta X

定理#

f(X)=(f1(X),...,fm(X))f(X)=(f_1(X),...,f_m(X))在X=X0X=X_0可微，等价于，每个fi(X)f_i(X)在X=X0X=X_0处可微
5.5.2 二元向量值函数 f(x1,x2)=(f1(x1,x2),...,fm(x1,x2))f(x_1,x_2)=(f_1(x_1,x_2),...,f_m(x_1,x_2))
若每个fif_i都在X0=(x0,1,x0,2)X_0=(x_{0,1},x_{0,2})可微，则称ff在X0X_{0}可微，也成ff在X0X_0可导可微，并称

d(f(X0))=(df1(X0),...,dfm(X0))T=[∂f1∂x1(X0)dx1+∂f1∂x2(X0)dx2...∂fm∂x1(X0)dx1+∂fm∂x2(X0)dx2]=[∂f1∂x1∂f1∂x2...∂fm∂x1∂fm∂x2][dx1dx2]\begin{align} d(f(X_0))&=(df_1(X_0),...,df_m(X_0))^T\\ &=\begin{bmatrix} \frac{\partial f_{1}}{\partial x_{1}}(X_0)dx_1+\frac{\partial f_{1}}{\partial x_{2}}(X_0)dx_2\\ ...\\ \frac{\partial f_{m}}{\partial x_{1}}(X_0)dx_1+\frac{\partial f_{m}}{\partial x_{2}}(X_0)dx_2\\ \end{bmatrix}\\ &=\begin{bmatrix} \frac{\partial f_{1}}{\partial x_{1}} & \frac{\partial f_{1}}{\partial x_{2}}\\ ...\\ \frac{\partial f_{m}}{\partial x_{1}}&\frac{\partial f_{m}}{\partial x_{2}}\\ \end{bmatrix}\begin{bmatrix} dx_1\\ dx_2 \end{bmatrix} \end{align}

为ff在X0X_0的微分，且称m×2m\times 2矩阵

A=[∂f1∂x1∂f1∂x2...∂fm∂x1∂fm∂x2]X=X0\begin{align} A=\begin{bmatrix} \frac{\partial f_{1}}{\partial x_{1}} & \frac{\partial f_{1}}{\partial x_{2}}\\ ...\\ \frac{\partial f_{m}}{\partial x_{1}}&\frac{\partial f_{m}}{\partial x_{2}}\\ \end{bmatrix}_{X=X_0} \end{align}

为ff在X0X_0的导数，此矩阵称为JacobiJacobi矩阵
此时df(X0)=A(dx1,dx2)T=∇df(X0)dXdf(X_0)=A(dx_1,d_x2)^T=\nabla df(X_0)dX
定理5.2 f:U(X0)⊂Rn→Rmf:U(X_0)\subset R^n\to R^m,ff在X0X_0可微的充分条件是每个∂fi∂xj\frac{\partial f_i}{\partial x_j}在X0X_0连续，1≤i≤m,1≤j≤n1\le i\le m,1\le j\le n

5.5.3 微分运算法则#

1.链式法则：#

设(u,v)→f(x1,x2)→g(y1,y2)(u,v)\overset{f}\to(x_1,x_2)\overset{g}\to(y1,y2)，都可微
则g∘fg\circ f也可微，且∇(g∘f)=∇(g)∇(f)\nabla (g\circ f)=\nabla(g)\nabla(f)

[∂y1∂u∂y1∂v∂y2∂u∂y2∂v]=[∂y1∂x1∂x1∂u+∂y1∂x2∂x2∂u∂y1∂x1∂x1∂v+∂y1∂x2∂x2∂v∂y2∂x1∂x1∂u+∂y2∂x2∂x2∂u∂y2∂x1∂x1∂v+∂y2∂x2∂x2∂v]=[∂y1∂x1∂y1∂x2∂y2∂x1∂y2∂x2][∂x1∂u∂x1∂v∂x2∂u∂x2∂v]\begin{align} \begin{bmatrix} \frac{\partial y_{1}}{\partial u}& \frac{\partial y_{1}}{\partial v}\\ \frac{\partial y_{2}}{\partial u}& \frac{\partial y_{2}}{\partial v}\\ \end{bmatrix} &= \begin{bmatrix} \frac{\partial y_{1}}{\partial x_1}\frac{\partial x_{1}}{\partial u}+\frac{\partial y_{1}}{\partial x_2}\frac{\partial x_{2}}{\partial u}&\frac{\partial y_{1}}{\partial x_1}\frac{\partial x_{1}}{\partial v}+\frac{\partial y_{1}}{\partial x_2}\frac{\partial x_{2}}{\partial v}\\ \frac{\partial y_{2}}{\partial x_1}\frac{\partial x_{1}}{\partial u}+\frac{\partial y_{2}}{\partial x_2}\frac{\partial x_{2}}{\partial u}&\frac{\partial y_{2}}{\partial x_1}\frac{\partial x_{1}}{\partial v}+\frac{\partial y_{2}}{\partial x_2}\frac{\partial x_{2}}{\partial v}\\ \end{bmatrix}\\ &=\begin{bmatrix} \frac{\partial y_{1}}{\partial x_{1}}& \frac{\partial y_{1}}{\partial x_{2}}\\ \frac{\partial y_{2}}{\partial x_{1}}& \frac{\partial y_{2}}{\partial x_{2}}\\ \end{bmatrix} \begin{bmatrix} \frac{\partial x_{1}}{\partial u}& \frac{\partial x_{1}}{\partial v}\\ \frac{\partial x_{2}}{\partial u}& \frac{\partial x_{2}}{\partial v}\\ \end{bmatrix} \end{align}

定理5.3 （4）f:R→R3,g:R→R3f:R\to R^{3},g:R\to R^3,都可微，则向量积f×gf\times g可微，且∇(f×g)=∇f×g+f×∇g\nabla (f\times g)=\nabla f\times g+f\times \nabla g#

proof:
设f=(f1,f2,f2),g=(g1,g2,g3)f=(f_1,f_2,f_2),g=(g_1,g_2,g_3),则

f×g=∣ijkf1f2f3g1g2g3∣=∣f2f3g2g3∣i+∣f3f1g3g1∣j+∣f1f2g1g2∣k\begin{align} f\times g= \begin{vmatrix} i&j&k\\ f_1&f_2&f_3\\ g_1&g_2&g_3\\ \end{vmatrix} =\begin{vmatrix} f_2&f_3\\ g_2&g_3\\ \end{vmatrix}i+\begin{vmatrix} f_3&f_1\\ g_3&g_1\\ \end{vmatrix}j+\begin{vmatrix} f_1&f_2\\ g_1&g_2\\ \end{vmatrix}k \end{align}∇(f×g)=∇∣f2f3g2g3∣i+∇∣f3f1g3g1∣j+∇∣f1f2g1g2∣k=∇(f2g3−f3g2)i+∇(f3g1−f1g3)j+∇(f1g2−f2g1)k=(f2′g3+f2g3′−f3′g2−f3g2′)i+(f3′g1+f3g1′−f1′g3−f1g3′)j+(f1′g2+f1g2′−f2′g1−f2g1′)k=(∣f2′f3′g2g3∣+∣f2f3g2′g3′∣)i+(∣f3′f1′g3g1∣+∣f3f1g3′g1′∣)j+(∣f1′f2′g1g2∣+∣f1f2g1′g2′∣)k=∣ijkf1′f2′f3′g1g2g3∣+∣ijkf1f2f3g1′g2′g3′∣=∇f×g+f×∇g\begin{align} \nabla(f\times g)&=\nabla \begin{vmatrix} f_2&f_3\\ g_2&g_3\\ \end{vmatrix}i+\nabla\begin{vmatrix} f_3&f_1\\ g_3&g_1\\ \end{vmatrix}j+\nabla\begin{vmatrix} f_1&f_2\\ g_1&g_2\\ \end{vmatrix}k\\ &=\nabla(f_2g_3-f_3g_2)i+\nabla(f_3g_1-f_1g_3)j+\nabla(f_1g_2-f_2g_1)k\\ &=(f_2'g_3+f_2g_3'-f_3'g_2-f_3g_2')i+(f_3'g_1+f_3g_1'-f_1'g_3-f_1g_3')j+(f_1'g_2+f_1g_2'-f_2'g_1-f_2g_1')k\\ &=(\begin{vmatrix} f_2'&f_3'\\ g_2&g_3\\ \end{vmatrix}+\begin{vmatrix} f_2&f_3\\ g_2'&g_3'\\ \end{vmatrix})i+(\begin{vmatrix} f_3'&f_1'\\ g_3&g_1\\ \end{vmatrix}+\begin{vmatrix} f_3&f_1\\ g_3'&g_1'\\ \end{vmatrix})j+(\begin{vmatrix} f_1'&f_2'\\ g_1&g_2\\ \end{vmatrix}+\begin{vmatrix} f_1&f_2\\ g_1'&g_2'\\ \end{vmatrix})k\\ &=\begin{vmatrix} i&j&k\\ f_1'&f_2'&f_3'\\ g_1&g_2&g_3\\ \end{vmatrix}+\begin{vmatrix} i&j&k\\ f_1&f_2&f_3\\ g_1'&g_2'&g_3'\\ \end{vmatrix}\\ &=\nabla f\times g+f\times \nabla g \end{align}

5.5.4 由方程组确定的隐函数组的微分方法#

设方程组F(x,y,u,v)=0,G(x,y,u,v)=0(5.36)F(x,y,u,v)=0,G(x,y,u,v)=0(5.36)
确定的两个定义在D⊂R2D\subset R^2上的函数u=u(x,y),v=v(x,y)u=u(x,y),v=v(x,y)
称为由(5.36)(5.36)确定的隐函数组
设F,G,u,vF,G,u,v都可微，对(5.36)(5.36)取微分得

{Fxdx+Fydy+Fudu+Fvdv=0Gxdx+Gydy+Gudu+Gvdv=0\begin{cases} F_xdx+F_ydy+F_udu+F_vdv=0\\ G_xdx+G_ydy+G_udu+G_vdv=0\\ \end{cases}(∗){Fudu+Fvdv=−Fxdx−FydyGudu+Gvdv=−Gxdx−Gydy(*) \begin{cases} F_udu+F_vdv=-F_xdx-F_ydy\\ G_udu+G_vdv=-G_xdx-G_ydy\\ \end{cases}

为了解出du,dvdu,dv
需要行列式

∂(F,G)∂(u,v)=∣FuFvGuGv∣≠0(5.37)\begin{align} \frac{\partial(F,G)}{\partial(u,v)}= \begin{vmatrix} F_u&F_v\\ G_u&G_v \end{vmatrix}\neq0(5.37) \end{align}

称为F,GF,G关于u,vu,v的JacobiJacobi行列式

定理5.5（隐函数组存在定理）#

(i)设F(x,y,u,v)F(x,y,u,v)与G(x,y,u,v)G(x,y,u,v)在区域V⊂R4V\subset R^4上连续，P0(x0,y0,u0,v0)P_0(x_0,y_0,u_0,v_0)为内点
(ii)F(P0)=G(P0)=0F(P_0)=G_(P_0)=0
(iii)F,GF,G在VV内有连续一阶偏导数
(iv)在P0P_0处，∂(F,G)∂(u,v)≠0\frac{\partial(F,G)}{\partial(u,v)}\neq0
则在点P0P_0的某个领域U(P0)U(P_0)内，方程组(5.36)(5.36)唯一确定一个定义在点Q0(x0,y0)Q_0(x_0,y_0)的某个领域U(Q0)U(Q_0)内的函数组u=f(x,y),v=g(x,y)u=f(x,y),v=g(x,y)使得
1∘,u0=f(x0,y0),v0=g(x0,y0)1^{\circ},u_0=f(x_0,y_0),v_0=g(x_0,y_0)
when(x,y)∈U(Q0)→(x,y,f(x,y),g(x,y)∈U(P0)),F(x,y,f(x,y),g(x,y))=0when (x,y)\in U(Q_0)\to(x,y,f(x,y),g(x,y)\in U(P_0)),F(x,y,f(x,y),g(x,y))=0
2∘2^{\circ}在U(Q0)U(Q_0)内，u=f(x,y),v=g(x,y)u=f(x,y),v=g(x,y)
有连续一阶偏导数，且du,dvdu,dv满足方程组(∗)(*)
即

{Fudu+Fvdv=−Fxdx−FydyGudu+Gvdv=−Gxdx−Gydy\begin{cases} F_udu+F_vdv=-F_xdx-F_ydy\\ G_udu+G_vdv=-G_xdx-G_ydy\\ \end{cases}

Ex1 讨论方程组#

(∗){F(x,y,u,v)=u2+v2−x2−y=0G(x,y,u,v)=−u+v−xy+1=0(*) \begin{cases} F(x,y,u,v)=u^2+v^2-x^2-y=0\\ G(x,y,u,v)=-u+v-xy+1=0 \end{cases}

在点P0(1,1,1,1)P_0(1,1,1,1)附近确定怎样的隐函数组，并求一阶微分
解
F(P0)=G(P0)=0F(P_0)=G(P_0)=0
F,GF,G在R4R^4内有一阶连续偏导数

(FuFvFxFyGuGvGxGy)P0=(2u2v−2x−1−11−y−x)P0=(22−2−1−11−1−1)\begin{align} \begin{pmatrix} F_u&F_v&F_x&F_y\\ G_u&G_v&G_x&G_y\\ \end{pmatrix}_{P_0} &=\begin{pmatrix} 2u&2v&-2x&-1\\ -1&1&-y&-x\\ \end{pmatrix}_{P_0}\\ &=\begin{pmatrix} 2&2&-2&-1\\ -1&1&-1&-1\\ \end{pmatrix} \end{align}

因为∂(F,G)∂(u,v)P0=4\frac{\partial(F,G)}{\partial(u,v)}_{P_0}=4所以确定u=u(x,y),v=v(x,y)u=u(x,y),v=v(x,y)
对(∗)(*)取微分得

{2udu+2vdv=2xdx+dy−du+dv=ydx+xdy\begin{cases} 2udu+2vdv&=2xdx+dy\\ -du+dv&=ydx+xdy \end{cases}

解出
du=(2x−2vy)dx+(1−2vx)dy2(u+v)du=\frac{(2x-2vy)dx+(1-2vx)dy}{2(u+v)}
dv=(2x+2vy)dx+(1+2ux)dy2(u+v)dv=\frac{(2x+2vy)dx+(1+2ux)dy}{2(u+v)}
改：求u,vu,v在点(1,1)(1,1)处的一阶微分
有

{2du+2dv=2dx+dy−du+dv=dx+dy\begin{cases} 2du+2dv&=2dx+dy\\ -du+dv&=dx+dy \end{cases}

5.6 多元微分学在几何上的简单应用#

5.6.1 空间曲线的切线与法平面#

1.曲线的参数方程#

平面曲线
x=x(t),y=y(t),α≤t≤βx=x(t),y=y(t),\alpha\le t\le\beta
空间曲线
x=x(t),y=y(t),z=z(t),α≤t≤βx=x(t),y=y(t),z=z(t),\alpha\le t\le\beta
r(t)=(x(t),y(t),z(t))r(t)=(x(t),y(t),z(t))

2.简单曲线与有向曲线#

若对于任意t1,t2∈(α,β),t1≠t2t_1,t_2\in(\alpha,\beta),t_1\neq t_2有r(t1)≠r(t2)r(t_1)\neq r(t_2)
即r:(α,β)→R3r:(\alpha,\beta)\to R^3是单射，曲线不自交，则称之为简单曲线
若还有r(α)=r(β)r(\alpha)=r(\beta)叫做简单闭曲线
对已知参数tt的曲线Γ\Gamma
规定tt增加的方向为正向，反之为负向
规定了方向的曲线称为有向曲线（定向曲线）
例如 xOyxOy平面内的圆周
r(t)=(cos⁡t,sin⁡t,0),t∈(0,2π)r(t)=(\cos t,\sin t,0),t\in(0,2\pi)
是正向曲线

3.空间曲线的切线与法平面#

（1）设空间简单曲线Γ\Gamma的参数方程为
r=r(t)=(x(t),y(t),z(t)),α≤t≤βr=r(t)=(x(t),y(t),z(t)),\alpha\le t\le\beta
其中r(t)r(t)在[α,β][\alpha,\beta]上可导，其导数记为
r′(t)=(x′(t),y′(t),z′(t)),r′(t)≠0r'(t)=(x'(t),y'(t),z'(t)),r'(t)\neq0
下面讨论Γ\Gamma在点P0(x(t0),y(t0),z(t0))P_0(x(t_0),y(t_0),z(t_0))处的切线，取动线P=r(t0+Δt)P=r(t_0+\Delta t)有各县P0PP_0P
当P→P0P\to P_0时，割线P0PP_0P的极限定义为Γ\Gamma在P0P_0的切线
割线P0PP_0P的方向
Δr=(Δx,Δy,Δz)=(x(t0+Δt)−x(t0),y(t0+Δt)−y(t0),z(t0+Δt)−z(t0))\Delta r=(\Delta x,\Delta y,\Delta z)=(x(t_0+\Delta t)-x(t_0),y(t_0+\Delta t)-y(t_0),z(t_0+\Delta t)-z(t_0))
ΔrΔt=(ΔxΔt,ΔyΔt,ΔzΔt)\frac{\Delta r}{\Delta t}=(\frac{\Delta x}{\Delta t},\frac{\Delta y}{\Delta t},\frac{\Delta z}{\Delta t})
也是P0PP_0P的方向
lim⁡Δt→0ΔrΔt=r′(t0)=(x′(t0),y′(t0),z′(t0))\lim_{\Delta t\to0 }\frac{\Delta r}{\Delta t}=r'(t_0)=(x'(t_0),y'(t_0),z'(t_0))
它指向切方向向量
参数tt指加的方向
对r=(x,y,z)r=(x,y,z)

dr=(dx,dy,dz)=(x′(t)dt,y′(t)dt,z′(t)dt)=(x′(t),y′(t),z′(t))dt=r′(t)dt\begin{align} dr&=(dx,dy,dz)\\ &=(x'(t)dt,y'(t)dt,z'(t)dt)\\ &=(x'(t),y'(t),z'(t))dt\\ &=r'(t)dt \end{align}

不一定指向tt增加的方向
因此，曲线Γ\Gamma在P0P_0的切线方程为
x−x0x′(t0)=y−y0y′(t0)=z−z0z′(t0)\frac{x-x_0}{x'(t_0)}=\frac{y-y_0}{y'(t_0)}=\frac{z-z_0}{z'(t_0)}
法平面过点P0P_0且垂直于切线，其方程为
x′(t0)(x−x0)+y′(t0)(y−y0)+z′(t0)(z−z0)=0x'(t_0)(x-x_0)+y'(t_0)(y-y_0)+z'(t_0)(z-z_0)=0

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