surface integral

vector field : a = (x,y,2) 

S : x^2+y^2+z^2=1, z>=0

 

solve int(S) a・ndS.

 

dSn・k = dxdy = zdS, dS=dxdy/z.

a・n = x^2+y^2+2z = 1-z^2+2z.

 

int(S) a・ndS = int(D) (-z + 2 + 1/z) dxdy, D={(x,y)|x^+y^2<=1}

= -2π/3 + 2π + 2π = 10π/3■

first integration : half volume of a sphere

second :  area of a circle

last : cartesian coordinates to polar coordinates