- #1
GwtBc
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- Homework Statement
- Let ## \omega = \frac{xdy -ydx}{x^2+y^2} ## and for ##p, q > 0## define the curve ##C_{p,q,x,y}##:
$$ (-p,-q) \rightarrow (x,-q) \rightarrow (x,y) $$
Let $$f(x,y) = \int_{C_{p,q,x,y}}\omega$$
Show that ##f(x,y)## is discontinuous at ##x=0## if ##y \geq 0## is fixed, but not if ## y < 0##. What is the significance of the value of this discontinuity?
- Relevant Equations
- Found the discontinuity to be ##2\pi## if ##y> 0## and ##\pi## if ##y=0##. However, I don't really know what the significance of this value is in this context.
after two simple line integrals we find that
$$ f(x,y) = \arctan{p/q} + \arctan{y/x} + \pi/2$$
if ##x > 0## and
$$f(x,y) = \arctan{p/q} + \arctan{y/x} - \pi/2$$
if ## x < 0 ##.
And then we can just take the limits to find the value of the discontinuity (as given above). But what is the significance of the value? I know it's a vague question but is there something very fundamental?
Perhaps something that's related is that earlier in the question there is discussion of how with ##\theta = \arctan(y/x)##, ##d\theta## is closed (as a differential form), but not exact.
$$ f(x,y) = \arctan{p/q} + \arctan{y/x} + \pi/2$$
if ##x > 0## and
$$f(x,y) = \arctan{p/q} + \arctan{y/x} - \pi/2$$
if ## x < 0 ##.
And then we can just take the limits to find the value of the discontinuity (as given above). But what is the significance of the value? I know it's a vague question but is there something very fundamental?
Perhaps something that's related is that earlier in the question there is discussion of how with ##\theta = \arctan(y/x)##, ##d\theta## is closed (as a differential form), but not exact.
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