Question

ques

Answer 1

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
\[[\phi]_{\phi(x,y_{1})}^{\phi(x,y_{2})}=\int\limits_{y_{1}}^{y_{2}} \frac{\partial \phi}{\partial y}dy+0\]\[\implies \int\limits_{y_{1}}^{y_{2}} \frac{\partial \phi}{\partial y}dy=\phi(x,y_{2})-\phi(x,y_{1})\]

Answer 2

FTC?

Answer 3

Seems like it.

Answer 4

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

Answer 5

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

Answer 6

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.

Answer 7

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

Answer 8

What if both x and y simultaneously varied?
\[\int\limits_{\phi(a,y_{1})}^{\phi(c,y_{2})}d\phi= \int\limits_{y_{1}}^{y_{2}} \frac{\partial \phi}{\partial y}dy+\int\limits_{a}^{c}\frac{\partial \phi}{\partial x}dx\]\[\phi(c,y_{2})-\phi(a,y_{1})=\phi(x,y_{2})-\phi(x,y_{1})+\phi(c,y)-\phi(a,y)\]
\[\phi(c,y_{2})-\phi(a,y_{1})=(\phi(x,y_{2})+\phi(c,y))-(\phi(x,y_{1}+\phi(a,y))=\phi_{\max}-\phi_{\min}\]

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

here's some background

Answer 13

i agree nish, this is really cool stuff!

Answer 14

Cheers!

Answer 15

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

Answer 16

@zzr0ck3r

lol!!! that is hilarious!!

Answer 17

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.

Answer 18

so
"What if both x and y simultaneously varied?"

you can't!