I derived the following equality using a few methods that aren't all that complicated. I'm wondering if there is any way to show this is true other than the way I used to derive it.
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I used a vector field based off of a gravitational field centered at zero:
F(x,y)=⟨x2+y2−x,x2+y2−y⟩
This field is conservative, so any closed path integral should result in zero. I decided to create an arbitrary path starting at (a,b) that follows a parabola to (a+2,b) then proceeds linearly back to (a,b). I set up the following parametric equations for these paths:
r1(t)=⟨t+a+1,t2+b−1⟩ for −1≤t≤1
r2(t)=⟨a+2−t,b⟩ for 0≤t≤2
The line integrals give us
∫r1F(x,y)dr+∫r2F(x,y)dr=0
∫r1F(x,y)ds=−∫r2F(x,y)dr
From here, if I have copied everything down correctly, it should give the claim above. I might elaborate on the last steps later.
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I used a vector field based off of a gravitational field centered at zero:
F(x,y)=⟨x2+y2−x,x2+y2−y⟩
This field is conservative, so any closed path integral should result in zero. I decided to create an arbitrary path starting at (a,b) that follows a parabola to (a+2,b) then proceeds linearly back to (a,b). I set up the following parametric equations for these paths:
r1(t)=⟨t+a+1,t2+b−1⟩ for −1≤t≤1
r2(t)=⟨a+2−t,b⟩ for 0≤t≤2
The line integrals give us
∫r1F(x,y)dr+∫r2F(x,y)dr=0
∫r1F(x,y)ds=−∫r2F(x,y)dr
From here, if I have copied everything down correctly, it should give the claim above. I might elaborate on the last steps later.