从源点(即电荷电流分布点)x'到场点x的距离r，以及矢径r分别为=sqrt ({(x-x))^2+((y-y'))^2+((z-z'))^2}-|||-=(x-x')(e)_(1)+(y-y'e)'+(z-z'e)e r=(x-x')ex+(y-y')ey+(z-z')ez 对源变数x'和场变数x求微商的算符分别为 =sqrt ({(x-x))^2+((y-y'))^2+((z-z'))^2}-|||-=(x-x')(e)_(1)+(y-y'e)'+(z-z'e)e=sqrt ({(x-x))^2+((y-y'))^2+((z-z'))^2}-|||-=(x-x')(e)_(1)+(y-y'e)'+(z-z'e)e

设r=√(x-x ) (}^2+{(y-y))^2+((z-z))^2为源点x到场点X的距离，r的方向规定为从源点指向场点。r=√(x-x ) (}^2+{(y

例1.3.1 已知 =(e)_(x)(x-x)+(e)_(y)(y-y)+(e)_(2)(z-z) ,R=|R| 。证-|||-明:-|||-(1) =dfra

( A ) = (x,y,z)|{x)^2+(y)^2+(z)^2=(a)^2,zgeqslant 0} ( B ) = (x,y,z)|{x)^2+(y)^

设 (x+y,x-y)=2((x)^2+(y)^2)(e)^(x^2-{y)^2}, 则 _(x)(x,y)-(f)_(y)(x,y)= __

(2) (x)^n-((x))^2+((x))^3=0-|||-(3) ^n+dfrac (2)(1-x)((x))^2=0 ;-|||-(4) +sqrt (

设(x,y,z)=(x)^2+(y)^3+z,求(x,y,z)=(x)^2+(y)^3+z,在点(x,y,z)=(x)^2+(y)^3+z,处沿方向(x,y,z

设 (x,y)=dfrac ({x)^2+(y)^2}({e)^xy+xysqrt ({x)^2+(y)^2}} ,则 (f)_(x)(1,0)= __ _.

6.微分方程 -2y+y=(x)^2(e)^x 的一个特解y"可设为 ()-|||-(A) ((b)_(0)(x)^2+(b)_(1)x)(x)^2(e)^x

[题目]设 (x)=ln (x+sqrt ({x)^2+(a)^2}) (a为常数),则-|||-(x)= __

12.设方程 +sqrt ({x)^2+(y)^2+(z)^2}=sqrt (2) 确定了函数 =z(x,y), 则z(x,y)-|||-在点 (1,0,-1)