Задача 54694 ...

5f3ea7e3faf909182968ddd9

20.10.2020 14:36:16

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z=x+iy
vector{z}=x-iy

[b]z*vector{z}[/b]=(x+iy)(x-iy)=x^2-(iy)^2=[b]x^2+y^2[/b]


f(z)=z*vector{z}=[b]x^2+y^2[/b]

u(x;y)=Ref(z)=[b]x^2+y^2[/b]
v(x;y)=Imf(z)=0

∫_(L) f(z)dz= ∫_(L) u(x;y)dx-v(x;y)dy+i* (∫_(L) v(x;y)dx+u(x;y)dy)


∫ _(AB) z*vector{z}dz= ∫_(AB)( x^2+y^2)dx-0*dy+i* (∫_(AB) 0*dx+(x^2+y^2)dy)=

АВ: y=3x -уравнение прямой АВ

x=t ⇒ dx=dt
y=3t ⇒ dy=3*dt
0 ≤ t ≤ 1

= ∫^(1)_(0) ( t^2+(3t)^2)dt-0*(3*dt)+i* (∫^(1)_(0) 0*dt+( t^2+(3t)^2)*(3*dt))=


= ∫^(1)_(0) 10t^2dt+i* (∫^(1)_(0) 30t^2dt)=

=10*(t^3/3)|^(1)_(0)+i*(30*(t^3/3))|^(1)_(0)=[b](10/3)+i*10[/b]