Let C be an arbitrary arc of the unit circle. Give a geometrical interpretation for ∫ C ▽ f ⋅ d r = 0 where f ( x , y ) = x 2 + y 2 .

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Let C be an arbitrary arc of the unit circle. Give a geometrical interpretation for  ∫  C  ▽  f  ⋅  d  r  =  0 where   f  (  x  ,  y  )  =      x    2    +      y    2    .

Lisa Acevedo · Accepted Answer

The integral of the scalar field   f with respect to arc length on the curve is      ∫    C    f  (  x  ,  y  )    d  sThis is never 0 unless the curve blackuces to a single point, because       x    2    +      y    2    &amp;gt;  0 except at the origin.But I suspect that&#039;s not the integral you&#039;re thinking of.Please clarify.EDIT: OK, so now you want      ∫    C    (  ∇  f  )  ⋅  d  rFor any continuously differentiable function   f, and any curve that starts at a point   a and ends at a point   b, that is   f  (  b  )  −  f  (  a  ). So in this case all you need for that to be 0 is that   f is the same at the starting and ending points of the curve, i.e. both are on the same circle centblack at the origin.

Let C be an arbitrary arc of the unit circle. Give a geometrical interpretation for ∫C∇f·dr= 0 where f(x,y)=x^2+y^2.

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