Proof for component test for conservative fields

Asked Dec 10 '23 at 15:10

Active Dec 10 '23 at 15:20

Viewed 33 times

The test for conservative fields is as follows:

The 2D vetor field is conservative if and only if:

$$∂_xF_y=∂_yF_x$$

$$∂_yF_x=∂_xF_y$$

The forward proof is quite straight forward to me, but I got stuck proving backwards.

I have tried using Green's Theorem:

from $∂_xF_y-∂_yF_x=0$, we can conclude that $\oint F_xdx+F_ydy=0$ for any closed loop, from here I got stuck but I thought about finding some special loop where x and y are held constant at some line on the loop so after integrating we can get some function whose partial derivatives are the x,y component of F, but I couldn't get any further. If someone can give me some help, it would be greatly appreciated.

edited Dec 10 '23 at 15:20

asked Dec 10 '23 at 15:10

user381761

When $F$ has no singularities then those partial differential relations imply by Green's theorem that the integral (i.e. work done against force $F$) is zero for any closed loop. Is that not what we call conservative? – Kurt G. Dec 10 '23 at 17:55
yes, but I mean how do we get from that to deducing F being a gradient of another scaler function? – user381761 Dec 12 '23 at 13:58
You are trying to prove the Poincare lemma. In your two dimensional case the link shows how to do it. It is pretty close to your own idea. – Kurt G. Dec 12 '23 at 14:15
oh, thanks, I didn't know it's already a lemma – user381761 Dec 12 '23 at 14:48
We are all standing on the shoulders of giants. You should also study the famous case when that lemma does not hold. – Kurt G. Dec 12 '23 at 14:55

Proof for component test for conservative fields

0 Answers0