【高等数学】多元微分学（一）

偏导数

偏导数定义

如果二元函数 f f f 在 x 0 , y 0 x_0,y_0 x0,y0 的某邻域有定义, 且下述极限存在
lim ⁡ Δ x → 0 f ( x 0 + Δ x , y 0 ) − f ( x 0 , y 0 ) Δ x \lim_{\Delta x\to 0} \frac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x} Δx→0limΔxf(x0+Δx,y0)−f(x0,y0)
其极限称作 f f f 关于 x x x 的偏导数, 记为

∂ f ∂ x ∣ ( x 0 , y 0 ) = lim ⁡ Δ x f ( x 0 + Δ x , y 0 ) − f ( x 0 , y 0 ) Δ x \frac{\partial f}{\partial x}|{(x_0,y_0)}=\lim{\Delta x} \frac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x} ∂x∂f∣(x0,y0)=ΔxlimΔxf(x0+Δx,y0)−f(x0,y0)

类似的
∂ f ∂ y ∣ ( x 0 , y 0 ) = lim ⁡ Δ y f ( x 0 , y 0 + Δ y ) − f ( x 0 , y 0 ) Δ y \frac{\partial f}{\partial y}|{(x_0,y_0)}= \lim{\Delta y} \frac{f(x_0,y_0+\Delta y)-f(x_0,y_0)}{\Delta y} ∂y∂f∣(x0,y0)=ΔylimΔyf(x0,y0+Δy)−f(x0,y0)

n n n 元函数的偏导数 u = f ( x 1 , ⋯ , x n ) u=f(x_1,\cdots,x_n) u=f(x1,⋯,xn),
∂ f ∂ x i = lim ⁡ Δ x i f ( x 1 , ⋯ , x i + Δ x i , ⋯ , x n ) − f ( x 1 , ⋯ , x i , ⋯ , x n ) Δ x i \frac{\partial f}{\partial x_i}= \lim_{\Delta x_i} \frac{f(x_1,\cdots, x_i+\Delta x_i, \cdots,x_n)-f(x_1,\cdots, x_i, \cdots, x_n)}{\Delta x_i} ∂xi∂f=ΔxilimΔxif(x1,⋯,xi+Δxi,⋯,xn)−f(x1,⋯,xi,⋯,xn)

导数性质 → \to → 偏导数性质

加 : ∂ ( f ( x , y ) + g ( x , y ) ) ∂ x = ∂ f ∂ x + ∂ g ∂ x 加:\frac{\partial (f(x,y)+g(x,y))}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial g}{\partial x} 加:∂x∂(f(x,y)+g(x,y))=∂x∂f+∂x∂g
减 : ∂ ( f ( x , y ) − g ( x , y ) ) ∂ x = ∂ f ∂ x − ∂ g ∂ x 减:\frac{\partial (f(x,y)-g(x,y))}{\partial x}=\frac{\partial f}{\partial x}-\frac{\partial g}{\partial x} 减:∂x∂(f(x,y)−g(x,y))=∂x∂f−∂x∂g
乘 : ∂ ( f ( x , y ) g ( x , y ) ) ∂ x = ∂ f ∂ x g + f ∂ g ∂ x 乘:\frac{\partial (f(x,y)g(x,y))}{\partial x}=\frac{\partial f}{\partial x} g+f\frac{\partial g}{\partial x} 乘:∂x∂(f(x,y)g(x,y))=∂x∂fg+f∂x∂g
除 : ∂ ( f ( x , y ) g ( x , y ) ) ∂ x = 1 g 2 ( ∂ f ∂ x g − f ∂ g ∂ x ) 除:\frac{\partial (\frac{f(x,y)}{g(x,y)})}{\partial x}=\frac{1}{g^2}\left(\frac{\partial f}{\partial x}g-f\frac{\partial g}{\partial x}\right) 除:∂x∂(g(x,y)f(x,y))=g21(∂x∂fg−f∂x∂g)

高阶偏导数

∂ 2 f ∂ x 2 = ∂ ∂ x ( ∂ f ∂ x ) \frac{\partial^2 f}{\partial x^2}= \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) ∂x2∂2f=∂x∂(∂x∂f)
∂ 2 f ∂ x ∂ y = ∂ ∂ y ( ∂ f ∂ x ) \frac{\partial^2 f}{\partial x\partial y}= \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) ∂x∂y∂2f=∂y∂(∂x∂f)
∂ 2 f ∂ y ∂ x = ∂ ∂ x ( ∂ f ∂ y ) \frac{\partial^2 f}{\partial y\partial x}= \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) ∂y∂x∂2f=∂x∂(∂y∂f)
∂ 2 f ∂ y 2 = ∂ ∂ y ( ∂ f ∂ y ) \frac{\partial^2 f}{\partial y^2}= \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) ∂y2∂2f=∂y∂(∂y∂f)

当 ∂ 2 f ∂ y ∂ x \frac{\partial^2 f}{\partial y\partial x} ∂y∂x∂2f, ∂ 2 f ∂ x ∂ y \frac{\partial^2 f}{\partial x\partial y} ∂x∂y∂2f 是连续函数时, ∂ 2 f ∂ y ∂ x = ∂ 2 f ∂ x ∂ y \frac{\partial^2 f}{\partial y\partial x}=\frac{\partial^2 f}{\partial x\partial y} ∂y∂x∂2f=∂x∂y∂2f.

全微分

u = f ( x , y ) u=f(x,y) u=f(x,y), $\Delta u= f(x+\Delta x, y+\Delta y)-f(x,y) $

定义存在 A , B A,B A,B, Δ z = A Δ x + B Δ y + o ( ρ ) \Delta z= A\Delta x+B\Delta y+ o(\rho) Δz=AΔx+BΔy+o(ρ), ρ = ( Δ x ) 2 + ( Δ y ) 2 \rho=\sqrt{(\Delta x)^2+(\Delta y)^2} ρ=(Δx)2+(Δy)2 , 称函数 f f f 在 ( x , y ) (x,y) (x,y) 处可微, d z d z dz 称为全微分, ( A , B ) (A,B) (A,B) 称为梯度.

当函数 f f f 可微时, d z = ∂ f ∂ x d x + ∂ f ∂ y d y dz= \frac{\partial f}{\partial x} dx+ \frac{\partial f}{\partial y}dy dz=∂x∂fdx+∂y∂fdy, 梯度计算公式为 ∇ f ( x , y ) = ( ∂ f ∂ x , ∂ f ∂ y ) ⊤ \nabla f(x,y)=(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})^\top ∇f(x,y)=(∂x∂f,∂y∂f)⊤

微分性质 → \to → 全微分性质

可微函数:
加 : d ( f + g ) = d f + d g 加: d(f+g)=df+dg 加:d(f+g)=df+dg
减 : d ( f − g ) = d f − d g 减: d(f-g)=df-dg 减:d(f−g)=df−dg
乘 : d ( f g ) = g d f + f d g 乘: d(fg)= g df+f dg 乘:d(fg)=gdf+fdg
除 : d ( f g ) = g d f − f d g g 2 除:d\left(\frac{f}{g}\right)=\frac{g df- f dg}{g^2} 除:d(gf)=g2gdf−fdg

偏导数性质 → \to → 梯度性质

可微函数:
加 : ∇ ( f + g ) = ∇ f + ∇ g 加: \nabla (f+g)=\nabla f+\nabla g 加:∇(f+g)=∇f+∇g
减 : ∇ ( f − g ) = ∇ f − ∇ g 减: \nabla (f-g)=\nabla f-\nabla g 减:∇(f−g)=∇f−∇g
乘 : ∇ ( f g ) = g ∇ f + f ∇ g 乘: \nabla (fg)= g \nabla f+f \nabla g 乘:∇(fg)=g∇f+f∇g
除 : ∇ ( f g ) = g ∇ f − f ∇ g g 2 除:\nabla \left(\frac{f}{g}\right)=\frac{g \nabla f- f \nabla g}{g^2} 除:∇(gf)=g2g∇f−f∇g

当 ∂ f ∂ x \frac{\partial f}{\partial x} ∂x∂f, ∂ f ∂ y \frac{\partial f}{\partial y} ∂y∂f 是连续函数时, f ( x , y ) f(x,y) f(x,y) 可微.

复合函数的微分法

双层复合偏导

一元内嵌一元函数 (全导数)

d d x f ( u ( x ) ) = d f d u d u d x \frac{d }{d x}f(u(x)) =\frac{d f}{d u}\frac{d u}{d x} dxdf(u(x))=dudfdxdu

f u x

一元内嵌二元函数

∂ ∂ x f ( u ( x , y ) ) = d f d u ∂ u ∂ x \frac{\partial }{\partial x}f(u(x,y)) =\frac{d f}{d u}\frac{\partial u}{\partial x} ∂x∂f(u(x,y))=dudf∂x∂u
∂ ∂ y f ( u ( x , y ) ) = d f d u ∂ u ∂ y \frac{\partial }{\partial y}f(u(x,y)) =\frac{d f}{d u}\frac{\partial u}{\partial y} ∂y∂f(u(x,y))=dudf∂y∂u

f u x y

一元内嵌三元函数

∂ ∂ x f ( u ( x , y , z ) ) = d f d u ∂ u ∂ x \frac{\partial }{\partial x}f(u(x,y,z)) =\frac{d f}{d u}\frac{\partial u}{\partial x} ∂x∂f(u(x,y,z))=dudf∂x∂u
∂ ∂ y f ( u ( x , y , z ) ) = d f d u ∂ u ∂ y \frac{\partial }{\partial y}f(u(x,y,z)) =\frac{d f}{d u}\frac{\partial u}{\partial y} ∂y∂f(u(x,y,z))=dudf∂y∂u
∂ ∂ z f ( u ( x , y , z ) ) = d f d u ∂ u ∂ z \frac{\partial }{\partial z}f(u(x,y,z)) =\frac{d f}{d u}\frac{\partial u}{\partial z} ∂z∂f(u(x,y,z))=dudf∂z∂u

f u x y z

二元内嵌一元函数 (全导数)

∂ ∂ x f ( u ( x ) , v ( x ) ) = ∂ f ∂ u d u d x + ∂ f ∂ v d v d x \frac{\partial }{\partial x}f(u(x),v(x)) =\frac{\partial f}{\partial u}\frac{d u}{d x}+\frac{\partial f}{\partial v}\frac{d v}{d x} ∂x∂f(u(x),v(x))=∂u∂fdxdu+∂v∂fdxdv

f u v x

二元内嵌二元函数

∂ ∂ x f ( u ( x , y ) , v ( x , y ) ) = ∂ f ∂ u ∂ u ∂ x + ∂ f ∂ v ∂ v ∂ x \frac{\partial }{\partial x}f(u(x,y),v(x,y)) =\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x} ∂x∂f(u(x,y),v(x,y))=∂u∂f∂x∂u+∂v∂f∂x∂v
∂ ∂ y f ( u ( x , y ) , v ( x , y ) ) = ∂ f ∂ u ∂ u ∂ y + ∂ f ∂ v ∂ v ∂ y \frac{\partial }{\partial y}f(u(x,y),v(x,y)) =\frac{\partial f}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial y} ∂y∂f(u(x,y),v(x,y))=∂u∂f∂y∂u+∂v∂f∂y∂v

f u v x y

二元内嵌三元函数

∂ ∂ x f ( u ( x , y , z ) , v ( x , y , z ) ) = ∂ f ∂ u ∂ u ∂ x + ∂ f ∂ v ∂ v ∂ x \frac{\partial }{\partial x}f(u(x,y,z),v(x,y,z)) =\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x} ∂x∂f(u(x,y,z),v(x,y,z))=∂u∂f∂x∂u+∂v∂f∂x∂v
∂ ∂ y f ( u ( x , y , z ) , v ( x , y , z ) ) = ∂ f ∂ u ∂ u ∂ y + ∂ f ∂ v ∂ v ∂ y \frac{\partial }{\partial y}f(u(x,y,z),v(x,y,z)) =\frac{\partial f}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial y} ∂y∂f(u(x,y,z),v(x,y,z))=∂u∂f∂y∂u+∂v∂f∂y∂v
∂ ∂ z f ( u ( x , y , z ) , v ( x , y , z ) ) = ∂ f ∂ u ∂ u ∂ z + ∂ f ∂ v ∂ v ∂ z \frac{\partial }{\partial z}f(u(x,y,z),v(x,y,z)) =\frac{\partial f}{\partial u}\frac{\partial u}{\partial z}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial z} ∂z∂f(u(x,y,z),v(x,y,z))=∂u∂f∂z∂u+∂v∂f∂z∂v

f u v w x y

三元内嵌一元函数 (全导数)

∂ ∂ x f ( u ( x ) , v ( x ) , w ( x ) ) = ∂ f ∂ u d u d x + ∂ f ∂ v d v d x + ∂ f ∂ w d w d x \frac{\partial }{\partial x}f(u(x),v(x),w(x)) =\frac{\partial f}{\partial u}\frac{d u}{d x}+\frac{\partial f}{\partial v}\frac{d v}{d x}+\frac{\partial f}{\partial w}\frac{d w}{d x} ∂x∂f(u(x),v(x),w(x))=∂u∂fdxdu+∂v∂fdxdv+∂w∂fdxdw

f u v w x

三元内嵌二元函数

∂ ∂ x f ( u ( x , y ) , v ( x , y ) , w ( x , y ) ) = ∂ f ∂ u ∂ u ∂ x + ∂ f ∂ v ∂ v ∂ x + ∂ f ∂ w ∂ w ∂ x \frac{\partial }{\partial x}f(u(x,y),v(x,y),w(x,y)) =\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x}+\frac{\partial f}{\partial w}\frac{\partial w}{\partial x} ∂x∂f(u(x,y),v(x,y),w(x,y))=∂u∂f∂x∂u+∂v∂f∂x∂v+∂w∂f∂x∂w
∂ ∂ y f ( u ( x , y ) , v ( x , y ) , w ( x , y ) ) = ∂ f ∂ u ∂ u ∂ y + ∂ f ∂ v ∂ v ∂ y + ∂ f ∂ w ∂ w ∂ y \frac{\partial }{\partial y}f(u(x,y),v(x,y),w(x,y)) =\frac{\partial f}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial y}+\frac{\partial f}{\partial w}\frac{\partial w}{\partial y} ∂y∂f(u(x,y),v(x,y),w(x,y))=∂u∂f∂y∂u+∂v∂f∂y∂v+∂w∂f∂y∂w