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Let $w$ be a differential form defined by $w(x,y)=xy^ndx+x^mydy$ where $m$ and $n$ are non negative integers.

1) For which values of $m$ and $n$ the differential form $w$ is closed?

2) For these values, is $w$ exact ? if yes then determine all of its primitives?

My try:

1) $w$ is defined for all couples $(x,y)$ such that $x>0$ and $y>0$. Moreover if $w$ is closed then necessarely $\dfrac{\partial (xy^n)}{\partial y}= \dfrac{\partial (x^my)}{\partial x}$ which means $nxy^{n-1}=mx^{m-1}y$ for all $x>0$ and $y>0$. In particular this identity must hold for $x=y=1$, in which case we must have $n=m$. Now this criterion gives that $nxy^{n-1}=nx^{n-1}y$ for all $x>0$ and $y>0$. If $n=m=0$, $w(x,y)=xdx+ydy$ is clearly closed. If $n=m\not = 0$, we have that $xy^{n-1}=x^{n-1}y$ for all $x>0$ and $y>0$, hence $ x^{n-2}=y^{n-2}$ for all $x>0$ and $y>0$, which can not be true. Hence the only value is $n=0$.

2)Since $w$ is defined on the subset $U$ of the plane consisting of couples $(x,y)$ such that $x>0$ and $y>0$ it is clear that $U$ is convex hence star convex and by Poincaré theorem, for $n=0$, $w$ is exact.

to find the primitives $f$ of $w$ we solve the equation $\dfrac{\partial f}{\partial x}=xy^0=x$ which gives that $f(x,y)=x^2/2+c_1(y)$ and the equation $\dfrac{\partial f}{\partial y}=x^0y=y$ which gives that $f(x,y)=y^2/2+c_2(x)$ Hence $f(x,y)=x^2/2+y^2/2+constant$.

Is my try correct? thank you for your help!

palio
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1 Answers1

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Everything you write is correct but for a minor point. When you say that $w(x,y)$ is defined only in the first quadrant, why so? You can of course dedice that it is only to be defined there, but its natural domain is the whole of $\mathbb{R}^2$ - it is not a problem if the function coefficient of $dx$ or $dy$ or even both become zero, they only need not to be singular. To be slightly more precise, if you have a differential 1-form in an $n$-dimensional manifold locally of the form $w=\sum_{i=1}^n f_i(x_1,\ldots,x_n)\mathrm{d}x^i$ then you want the functions $f_i(x_1,\ldots,x_n)$ to satisfy some regularity condition - typically to be $C^{\infty}$ in smooth manifold theory, but you could ask them to be conitnuous, $C^2$, or whatever, but it is generally not a problem if they vanish at some points.

GFR
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