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anonymous
 one year ago
ques
anonymous
 one year ago
ques

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Consider a region R covered from x=a to x=c and y=y1(x) to y=y2(x) let \[\frac{\partial \phi(x,y)}{\partial y}=\frac{\partial \phi}{\partial y}\] be continuous on it Then, \[\int\limits_{y_{1}}^{y_{2}} \frac{\partial \phi }{\partial y}dy=\phi(x,y_{2})\phi(x,y_{1})\] But how?? Here is what I think is how it's supposed to be.. \[d\phi=\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial x}dx\] Integrating while keeping x constant we get \[\int\limits_{\phi(x,y_{1})}^{\phi(x,y_{2})}d\phi=\int\limits_{y_{1}}^{y_{2}}\frac{\partial \phi}{\partial y}.dy+\int\limits_{x}^{x} \frac{\partial \phi}{\partial x}dx\] Now \[\int\limits_{a}^{a}f(x)dx=0\] so we get \[[\phi]_{\phi(x,y_{1})}^{\phi(x,y_{2})}=\int\limits_{y_{1}}^{y_{2}} \frac{\partial \phi}{\partial y}dy+0\]\[\implies \int\limits_{y_{1}}^{y_{2}} \frac{\partial \phi}{\partial y}dy=\phi(x,y_{2})\phi(x,y_{1})\]

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.0I would like a solid example of this so i can picture how phi is moving though. :\

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Similarly we can get \[\int\limits_{a}^{c} \frac{\partial \phi}{\partial x}dx=\phi(c,y)\phi(a,y)\]

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.3this is a really powerful statement, you made it right at the start \[\int\limits_{y_{1}}^{y_{2}} \frac{\partial \phi }{\partial y}dy=\phi(x,y_{2})\phi(x,y_{1})\] here, you are taking a partial in one term and generating a potential function for a conservative field, all in one go.

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.3and one you have a conservative potential, you can pretty much switch your brain off

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0What if both x and y simultaneously varied? \[\int\limits_{\phi(a,y_{1})}^{\phi(c,y_{2})}d\phi= \int\limits_{y_{1}}^{y_{2}} \frac{\partial \phi}{\partial y}dy+\int\limits_{a}^{c}\frac{\partial \phi}{\partial x}dx\]\[\phi(c,y_{2})\phi(a,y_{1})=\phi(x,y_{2})\phi(x,y_{1})+\phi(c,y)\phi(a,y)\] \[\phi(c,y_{2})\phi(a,y_{1})=(\phi(x,y_{2})+\phi(c,y))(\phi(x,y_{1}+\phi(a,y))=\phi_{\max}\phi_{\min}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Interesting stuff, the 1st post is actually part of Green's Theorem

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.0Now I see a relation t that. Im understanding your forms, but not so much of whats going on in the problem.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I wanted to know how I arrived at \[\int\limits_{y_{1}}^{y_{2}}\frac{\partial \phi}{\partial y}dy=\phi(x,y_{2})\phi(x,y_{1})\] was correct

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.3here's some background

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.3i agree nish, this is really cool stuff!

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.0I hate this stuff :) I like my math to be completely useless.

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.3@zzr0ck3r lol!!! that is hilarious!!

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.3i'm wondering, what does this actually mean?! \[\int\limits_{\phi(a,y_{1})}^{\phi(c,y_{2})}d\phi= \int\limits_{y_{1}}^{y_{2}} \frac{\partial \phi}{\partial y}dy+\int\limits_{a}^{c}\frac{\partial \phi}{\partial x}dx\] looks like a line integral, but the limits remind me of a area integral, where you actually do an iterated integral or an area, as opposed to a line integral between distinct points or along a distinct path and line integrals are integrals in 1 variable only, always. so you paramaterise or you find the relationship between say, x and y, and you build that into the 1 integration in 1 variable.

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.3so "What if both x and y simultaneously varied?" you can't!
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