vector field's divergence and flux?

Compute the 2 Dimensional divergence of the vector field and use green's theorem, flux form to evaluate the integral

F=<y,-x>

R is a square with the verticies (0,0), (1,0), (1,1), and (0,1).

Divergence test eq.-

F=<f,g>

∂f/∂x + ∂g/∂y

Greens theorem, flux form eq-

F=<f,g>

∮F∙n ds = ∮f*dy-g*dx = ∬_R( ∂f/∂x+∂g/∂y)dA

Update:

I don't know understand this concept at all. Please show your work. Thank you.

1 Answer

Relevance
  • 9 years ago
    Favorite Answer

    ∮c F·dr = ∫∫ ∇ x F · n dS ............. Green's in the x-y plane

    ∇ x F = ∂/∂x [-x] - ∂/∂y [y] = {0,0,-2}

    ... n dS = <0,0,1> dx dy for positively oriented curves the x-y plane

    = ∫∫ (-2) dx dy

    = -2 ....... ∫∫ dx dy = 1 for a unit square

    Answer: -2

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