- #1
PhDeezNutz
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- TL;DR Summary
- As the title of the thread states. A set of notes https://www.math.purdue.edu/~arapura/preprints/diffforms.pdf#page4
From section 1.4 (pages 4-7)
Theorem 1.4.1 states that if a closed differential 1-form on ##R^2## must be exact (i.e. a total differential of scalar function ##f \left( x,y \right)##)
I don’t understand how the proof provided is valid
A differential form on ##R^2## has the following form
##df = Fdx + Gdy##
It is closed if ##\frac{\partial F}{\partial y} = \frac{\partial G}{\partial x}##
And like I said earlier, a form is exact if there exists scalar function ##f \left(x,y\right)## such that ##F = \frac{df}{dx}## and ##G = \frac{df}{dy}##
In order to prove closed ##\Rightarrow## exact the author does a line integral along two joined paths.
1. The path from ##(0,0)## to ##(x,0)## where ##x## changes and ##y## is constant
Unioned with
2. The from ##(x,0)## to ##(x,y)## where ##y## changes and ##x## is constant
They seem to integrate and then re-differentiate as follows
##df = F dx + G dy##
## \int \, df = \int F \,dx + \int G \,dy##
##f = \int_{0}^{x} F\left(t,0 \right) \,dt + \int_{0}^{y} G\left(x,t\right) \,dt##
If we take the partial ##x## and partial ##y## derivative using the fundamental theorem of calculus we very much get
##\frac{\partial f}{\partial x} = F \left(x,y\right)##
##\frac{\partial f}{\partial y} = G \left(x,y\right)##
This is the sketch of the “proof” but I don’t understand where the assumption of closedness was used. Or any related corollary.
Can someone please guide me? Did I get the definition of closed wrong?
##df = Fdx + Gdy##
It is closed if ##\frac{\partial F}{\partial y} = \frac{\partial G}{\partial x}##
And like I said earlier, a form is exact if there exists scalar function ##f \left(x,y\right)## such that ##F = \frac{df}{dx}## and ##G = \frac{df}{dy}##
In order to prove closed ##\Rightarrow## exact the author does a line integral along two joined paths.
1. The path from ##(0,0)## to ##(x,0)## where ##x## changes and ##y## is constant
Unioned with
2. The from ##(x,0)## to ##(x,y)## where ##y## changes and ##x## is constant
They seem to integrate and then re-differentiate as follows
##df = F dx + G dy##
## \int \, df = \int F \,dx + \int G \,dy##
##f = \int_{0}^{x} F\left(t,0 \right) \,dt + \int_{0}^{y} G\left(x,t\right) \,dt##
If we take the partial ##x## and partial ##y## derivative using the fundamental theorem of calculus we very much get
##\frac{\partial f}{\partial x} = F \left(x,y\right)##
##\frac{\partial f}{\partial y} = G \left(x,y\right)##
This is the sketch of the “proof” but I don’t understand where the assumption of closedness was used. Or any related corollary.
Can someone please guide me? Did I get the definition of closed wrong?