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向量分析 向量场的微分与积分:散度、旋度,以及将它们联系起来的积分定理。
直观 旋转又扩散的场:散度与旋度的直觉 设想空间中每一点水流的速度:这正是一个向量场 F ( x , y , z ) \mathbf{F}(x,y,z) F ( x , y , z ) F \mathbf{F} F 散度 ∇ ⋅ F \nabla \cdot \mathbf{F} ∇ ⋅ F 旋度 ∇ × F \nabla \times \mathbf{F} ∇ × F
一个鞍形势函数 φ ( x , y ) = x 2 − y 2 \varphi(x,y) = x^2 - y^2 φ ( x , y ) = x 2 − y 2 F = ∇ φ = ( 2 x , − 2 y ) \mathbf{F} = \nabla \varphi = (2x, -2y) F = ∇ φ = ( 2 x , − 2 y ) 0 0 0 0 0 0 大学 严格定义:散度、旋度与边界积分定理 定义: 向量场的散度
对于 R 3 \mathbb{R}^3 R 3 F = ( F 1 , F 2 , F 3 ) \mathbf{F} = (F_1, F_2, F_3) F = ( F 1 , F 2 , F 3 ) ∇ ⋅ F \nabla \cdot \mathbf{F} ∇ ⋅ F div F ( p ) > 0 \operatorname{div}\mathbf{F}(p) > 0 div F ( p ) > 0 p p p div F ( p ) < 0 \operatorname{div}\mathbf{F}(p) < 0 div F ( p ) < 0
∇ ⋅ F = ∂ F 1 ∂ x + ∂ F 2 ∂ y + ∂ F 3 ∂ z \nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} ∇ ⋅ F = ∂ x ∂ F 1 + ∂ y ∂ F 2 + ∂ z ∂ F 3 旋度 ∇ × F \nabla \times \mathbf{F} ∇ × F ∇ × F ( p ) \nabla \times \mathbf{F}(p) ∇ × F ( p ) ∣ ∇ × F ( p ) ∣ |\nabla \times \mathbf{F}(p)| ∣∇ × F ( p ) ∣ ∇ × F = 0 \nabla \times \mathbf{F} = \mathbf{0} ∇ × F = 0 无旋场 (若定义在单连通区域上,也称为保守场),它恰好就是作为某个标量势 φ \varphi φ F = ∇ φ \mathbf{F} = \nabla \varphi F = ∇ φ
∇ × F = ( ∂ F 3 ∂ y − ∂ F 2 ∂ z , ∂ F 1 ∂ z − ∂ F 3 ∂ x , ∂ F 2 ∂ x − ∂ F 1 ∂ y ) \nabla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z},\ \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x},\ \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) ∇ × F = ( ∂ y ∂ F 3 − ∂ z ∂ F 2 , ∂ z ∂ F 1 − ∂ x ∂ F 3 , ∂ x ∂ F 2 − ∂ y ∂ F 1 ) 向量分析的三个边界积分定理 定理 区域及其边界 积分恒等式 格林定理 平面区域 D D D ∂ D \partial D ∂ D ∮ ∂ D ( P d x + Q d y ) = ∬ D ( ∂ Q ∂ x − ∂ P ∂ y ) d A \oint_{\partial D} (P\,dx + Q\,dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA ∮ ∂ D ( P d x + Q d y ) = ∬ D ( ∂ x ∂ Q − ∂ y ∂ P ) d A 斯托克斯定理 空间中的曲面 S S S ∂ S \partial S ∂ S ∮ ∂ S F ⋅ d r = ∬ S ( ∇ × F ) ⋅ d S \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} ∮ ∂ S F ⋅ d r = ∬ S ( ∇ × F ) ⋅ d S 散度定理(高斯定理) 空间中的立体区域 V V V ∂ V \partial V ∂ V ∬ ∂ V F ⋅ d S = ∭ V ( ∇ ⋅ F ) d V \iint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F})\,dV ∬ ∂ V F ⋅ d S = ∭ V ( ∇ ⋅ F ) d V
大学 核心定理:格林定理与散度定理 设 D ⊂ R 2 D \subset \mathbb{R}^2 D ⊂ R 2 ∂ D \partial D ∂ D P ( x , y ) P(x,y) P ( x , y ) Q ( x , y ) Q(x,y) Q ( x , y ) D D D ∮ ∂ D ( P d x + Q d y ) = ∬ D ( ∂ Q ∂ x − ∂ P ∂ y ) d A \oint_{\partial D} (P\,dx + Q\,dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA ∮ ∂ D ( P d x + Q d y ) = ∬ D ( ∂ x ∂ Q − ∂ y ∂ P ) d A
为什么成立? 右边把局部的「环量密度」∂ Q ∂ x − ∂ P ∂ y \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ∂ x ∂ Q − ∂ y ∂ P D D D ∂ D \partial D ∂ D
证明 首先对矩形 R = [ a , b ] × [ c , d ] R = [a,b]\times[c,d] R = [ a , b ] × [ c , d ] D = R D = R D = R ∬ R ∂ Q ∂ x d A = ∫ c d ( Q ( b , y ) − Q ( a , y ) ) d y \iint_R \frac{\partial Q}{\partial x}\,dA = \int_c^d \left(Q(b,y) - Q(a,y)\right) dy ∬ R ∂ x ∂ Q d A = ∫ c d ( Q ( b , y ) − Q ( a , y ) ) d y ∬ R ∂ P ∂ y d A = ∫ a b ( P ( x , d ) − P ( x , c ) ) d x \iint_R \frac{\partial P}{\partial y}\,dA = \int_a^b \left(P(x,d) - P(x,c)\right) dx ∬ R ∂ y ∂ P d A = ∫ a b ( P ( x , d ) − P ( x , c ) ) d x
这两式恰好分别是 Q d y Q\,dy Q d y R R R P d x P\,dx P d x ∬ R ( ∂ Q ∂ x − ∂ P ∂ y ) d A = ∮ ∂ R ( P d x + Q d y ) \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA = \oint_{\partial R} (P\,dx + Q\,dy) ∬ R ( ∂ x ∂ Q − ∂ y ∂ P ) d A = ∮ ∂ R ( P d x + Q d y )
对于一般区域 D D D D D D
抵消之后剩下的正是沿外边界 ∂ D \partial D ∂ D ∬ D ( ∂ Q ∂ x − ∂ P ∂ y ) d A \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA ∬ D ( ∂ x ∂ Q − ∂ y ∂ P ) d A D D D
设 V ⊂ R 3 V \subset \mathbb{R}^3 V ⊂ R 3 ∂ V \partial V ∂ V F \mathbf{F} F V V V ∬ ∂ V F ⋅ d S = ∭ V ( ∇ ⋅ F ) d V \iint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F})\,dV ∬ ∂ V F ⋅ d S = ∭ V ( ∇ ⋅ F ) d V
为什么成立? 与格林定理类似,散度的体积积分把 V V V ∂ V \partial V ∂ V
证明 首先对长方体 B = [ a 1 , b 1 ] × [ a 2 , b 2 ] × [ a 3 , b 3 ] B = [a_1,b_1]\times[a_2,b_2]\times[a_3,b_3] B = [ a 1 , b 1 ] × [ a 2 , b 2 ] × [ a 3 , b 3 ] V = B V = B V = B F = ( F 1 , F 2 , F 3 ) \mathbf{F} = (F_1, F_2, F_3) F = ( F 1 , F 2 , F 3 ) ∭ B ∂ F 1 ∂ x d V = ∬ ( F 1 ( b 1 , y , z ) − F 1 ( a 1 , y , z ) ) d y d z \iiint_B \frac{\partial F_1}{\partial x}\,dV = \iint \left(F_1(b_1,y,z) - F_1(a_1,y,z)\right) dy\,dz ∭ B ∂ x ∂ F 1 d V = ∬ ( F 1 ( b 1 , y , z ) − F 1 ( a 1 , y , z ) ) d y d z y y y z z z
每个右边恰好是 F \mathbf{F} F B B B ∭ B ( ∇ ⋅ F ) d V = ∬ ∂ B F ⋅ d S \iiint_B (\nabla \cdot \mathbf{F})\,dV = \iint_{\partial B} \mathbf{F} \cdot d\mathbf{S} ∭ B ( ∇ ⋅ F ) d V = ∬ ∂ B F ⋅ d S
对于一般的立体区域 V V V V V V
抵消之后剩下的通量恰好穿过外边界曲面 ∂ V \partial V ∂ V ∭ V ( ∇ ⋅ F ) d V \iiint_V (\nabla \cdot \mathbf{F})\,dV ∭ V ( ∇ ⋅ F ) d V V V V
大学 实际应用与典型例题 散度、旋度以及边界积分定理是经典场论的数学骨架:电磁学的麦克斯韦方程组 完全用 ∇ ⋅ E \nabla \cdot \mathbf{E} ∇ ⋅ E ∇ ⋅ B \nabla \cdot \mathbf{B} ∇ ⋅ B ∇ × E \nabla \times \mathbf{E} ∇ × E ∇ × B \nabla \times \mathbf{B} ∇ × B 流体力学 用散度定理导出质量守恒的连续性方程,并把不可压缩流动等同于 ∇ ⋅ F = 0 \nabla \cdot \mathbf{F} = 0 ∇ ⋅ F = 0 ∇ × F \nabla \times \mathbf{F} ∇ × F
例题: 在球面上验证径向场的散度定理
设 F ( x , y , z ) = ( x , y , z ) \mathbf{F}(x,y,z) = (x,y,z) F ( x , y , z ) = ( x , y , z ) S S S a a a x 2 + y 2 + z 2 = a 2 x^2+y^2+z^2=a^2 x 2 + y 2 + z 2 = a 2 V V V ∬ S F ⋅ d S = ∭ V ( ∇ ⋅ F ) d V \iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F})\,dV ∬ S F ⋅ d S = ∭ V ( ∇ ⋅ F ) d V
解答 先计算右边:∇ ⋅ F = ∂ x ∂ x + ∂ y ∂ y + ∂ z ∂ z = 1 + 1 + 1 = 3 \nabla \cdot \mathbf{F} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1+1+1 = 3 ∇ ⋅ F = ∂ x ∂ x + ∂ y ∂ y + ∂ z ∂ z = 1 + 1 + 1 = 3 ∭ V ( ∇ ⋅ F ) d V = 3 ⋅ vol ( V ) = 3 ⋅ 4 3 π a 3 = 4 π a 3 \iiint_V (\nabla \cdot \mathbf{F})\,dV = 3 \cdot \operatorname{vol}(V) = 3 \cdot \frac{4}{3}\pi a^3 = 4\pi a^3 ∭ V ( ∇ ⋅ F ) d V = 3 ⋅ vol ( V ) = 3 ⋅ 3 4 π a 3 = 4 π a 3
现在直接计算左边。在球面 S S S S S S a a a n = 1 a ( x , y , z ) \mathbf{n} = \frac{1}{a}(x,y,z) n = a 1 ( x , y , z ) F ⋅ n = 1 a ( x 2 + y 2 + z 2 ) = a 2 a = a \mathbf{F} \cdot \mathbf{n} = \frac{1}{a}(x^2+y^2+z^2) = \frac{a^2}{a} = a F ⋅ n = a 1 ( x 2 + y 2 + z 2 ) = a a 2 = a
由于 F ⋅ n = a \mathbf{F} \cdot \mathbf{n} = a F ⋅ n = a S S S ∬ S F ⋅ d S = a ⋅ area ( S ) = a ⋅ 4 π a 2 = 4 π a 3 \iint_S \mathbf{F} \cdot d\mathbf{S} = a \cdot \operatorname{area}(S) = a \cdot 4\pi a^2 = 4\pi a^3 ∬ S F ⋅ d S = a ⋅ area ( S ) = a ⋅ 4 π a 2 = 4 π a 3
例题: 利用格林定理计算面积
设 C C C x 2 + y 2 = 1 x^2+y^2=1 x 2 + y 2 = 1 P = − y P=-y P = − y Q = x Q=x Q = x D D D area ( D ) = 1 2 ∮ C ( x d y − y d x ) \operatorname{area}(D) = \frac{1}{2}\oint_C (x\,dy - y\,dx) area ( D ) = 2 1 ∮ C ( x d y − y d x )
解答 取 P = − y P=-y P = − y Q = x Q=x Q = x ∮ C ( − y d x + x d y ) = ∬ D ( ∂ x ∂ x − ∂ ( − y ) ∂ y ) d A = ∬ D ( 1 − ( − 1 ) ) d A = 2 ∬ D d A = 2 area ( D ) \oint_C (-y\,dx + x\,dy) = \iint_D \left(\frac{\partial x}{\partial x} - \frac{\partial(-y)}{\partial y}\right) dA = \iint_D (1 - (-1))\,dA = 2\iint_D dA = 2\operatorname{area}(D) ∮ C ( − y d x + x d y ) = ∬ D ( ∂ x ∂ x − ∂ y ∂ ( − y ) ) d A = ∬ D ( 1 − ( − 1 )) d A = 2 ∬ D d A = 2 area ( D )
将 C C C x = cos θ x=\cos\theta x = cos θ y = sin θ y=\sin\theta y = sin θ θ ∈ [ 0 , 2 π ] \theta \in [0, 2\pi] θ ∈ [ 0 , 2 π ] d x = − sin θ d θ dx = -\sin\theta\,d\theta d x = − sin θ d θ d y = cos θ d θ dy = \cos\theta\,d\theta d y = cos θ d θ ∮ C ( − y d x + x d y ) = ∫ 0 2 π ( sin 2 θ + cos 2 θ ) d θ = ∫ 0 2 π 1 d θ = 2 π \oint_C (-y\,dx + x\,dy) = \int_0^{2\pi} \left(\sin^2\theta + \cos^2\theta\right) d\theta = \int_0^{2\pi} 1\,d\theta = 2\pi ∮ C ( − y d x + x d y ) = ∫ 0 2 π ( sin 2 θ + cos 2 θ ) d θ = ∫ 0 2 π 1 d θ = 2 π
由于单位圆盘满足 area ( D ) = π \operatorname{area}(D) = \pi area ( D ) = π 2 area ( D ) = 2 π 2\operatorname{area}(D) = 2\pi 2 area ( D ) = 2 π 2 π 2\pi 2 π area ( D ) = 1 2 ∮ C ( x d y − y d x ) \operatorname{area}(D) = \frac{1}{2}\oint_C (x\,dy - y\,dx) area ( D ) = 2 1 ∮ C ( x d y − y d x )
常见错误. 两个常见错误:(1) 当向量场在所包围区域内部存在奇点时仍套用格林定理、斯托克斯定理或散度定理——例如 F = 1 x 2 + y 2 ( − y , x ) \mathbf{F} = \frac{1}{x^2+y^2}(-y,x) F = x 2 + y 2 1 ( − y , x ) ( 0 , 0 ) (0,0) ( 0 , 0 ) ∂ D \partial D ∂ D 历史注记
散度定理常被称为高斯定理 ,因为卡尔·弗里德里希·高斯在1813年研究牛顿引力下椭球体的引力时使用过它的一个版本。乔治·格林在1828年一本自费印刷、关于电学与磁学的小册子中提出了以他名字命名的平面版定理,但直到1850年代威廉·汤姆孙(开尔文勋爵)重新发现并再版之前,这一结果几乎无人问津。如今以乔治·加布里埃尔·斯托克斯命名的定理,据斯托克斯本人所说,最早是1850年开尔文在一封信中告诉他的,后来斯托克斯把它作为1854年剑桥史密斯奖考试的一道题目。
卡尔·弗里德里希·高斯
计算 F ( x , y , z ) = ( x 2 , y 2 , z 2 ) \mathbf{F}(x,y,z) = (x^2, y^2, z^2) F ( x , y , z ) = ( x 2 , y 2 , z 2 ) ( 1 , 1 , 1 ) (1,1,1) ( 1 , 1 , 1 ) ∇ ⋅ F \nabla \cdot \mathbf{F} ∇ ⋅ F
计算旋转场 F ( x , y , z ) = ( y , − x , 0 ) \mathbf{F}(x,y,z) = (y,-x,0) F ( x , y , z ) = ( y , − x , 0 ) ∇ × F \nabla \times \mathbf{F} ∇ × F
( 0 , 0 , − 2 ) (0,0,-2) ( 0 , 0 , − 2 ) ( 0 , 0 , 2 ) (0,0,2) ( 0 , 0 , 2 ) ( 2 , 0 , 0 ) (2,0,0) ( 2 , 0 , 0 ) ( 0 , 0 , 0 ) (0,0,0) ( 0 , 0 , 0 ) 在流体速度场 F \mathbf{F} F ∇ ⋅ F > 0 \nabla \cdot \mathbf{F} > 0 ∇ ⋅ F > 0
流体正被压缩向该点(净汇)。 该点是净源:向外流出的比流入的多。 流体绕该点做刚性旋转而不膨胀。 流体在该点恰好不可压缩。
哪个定理把开曲面 S S S ∇ × F \nabla \times \mathbf{F} ∇ × F ∂ S \partial S ∂ S
格林定理 散度定理 斯托克斯定理 微积分基本定理