use Green’s Theorem to find the counterclockwise circulation and outward flux for the field F...

Kye

Kye

Answered

2020-11-01

use Green’s Theorem to find the counterclockwise circulation and outward flux for the field F and the curve C. F=(y2x2)i+(x2+y2)j
C: The triangle bounded by y = 0, x = 3, and y = x

Answer & Explanation

sweererlirumeX

sweererlirumeX

Expert

2020-11-02Added 91 answers

Step 1
Given: F=(y2x2)i+(x2+y2)j
C: The triangle bounded by y=0, x=3, and y=x.
Step 2
Counterclockwise circulation:
To find counterclockwise circulation use, CFTds=R(NxMy)dxdy.
Here, F=(y2x2)i+(x2+y2)j
M=y2x2,N=x2+y2
My=2y,Nx=2x
We have {R0x3,0yx}.
Therefore, the counterclockwise circulation is given by
CFTds=030x(2x2y)dydx
=03[0x(2x2y)dy]dx
=03[2xy2y22]0xdx
=03[2xyy2]0xdx
=03[(2x2x2)0]dx
=03x2dx
=[x33]03
=(3)330
=9
Hence, the counterclockwise circulation = 9.
Step 3
Outward flux:
To find outward flux use, CFnds=R(Mx+Ny)dxdy.
Here, F=(y2x2)i+(x2+y2)j

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