## anonymous one year ago ques

1. anonymous

Consider 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})$

2. zzr0ck3r

FTC?

3. Jhannybean

Seems like it.

4. Jhannybean

I would like a solid example of this so i can picture how phi is moving though. :\

5. anonymous

Similarly we can get $\int\limits_{a}^{c} \frac{\partial \phi}{\partial x}dx=\phi(c,y)-\phi(a,y)$

6. IrishBoy123

this 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.

7. IrishBoy123

and one you have a conservative potential, you can pretty much switch your brain off

8. anonymous

What 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)$ $\phi(c,y_{2})-\phi(a,y_{1})=(\phi(x,y_{2})+\phi(c,y))-(\phi(x,y_{1}+\phi(a,y))=\phi_{\max}-\phi_{\min}$

9. anonymous

Interesting stuff, the 1st post is actually part of Green's Theorem

10. Jhannybean

Now I see a relation t that. Im understanding your forms, but not so much of whats going on in the problem.

11. anonymous

I 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

12. IrishBoy123

here's some background

13. IrishBoy123

i agree nish, this is really cool stuff!

14. anonymous

Cheers!

15. zzr0ck3r

I hate this stuff :) I like my math to be completely useless.

16. IrishBoy123

@zzr0ck3r lol!!! that is hilarious!!

17. IrishBoy123

i'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.

18. IrishBoy123

so "What if both x and y simultaneously varied?" you can't!