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Differential forms and cohomology are somewhat less intuitive than integration (at least for me), so maybe it is no easy to find such a neat example.

Anyway, let's try this one. Consider the 1-form

$\omega:=\frac{xdy-ydx}{x^2+y^2}$

in $X:=\mathbb{R}^2 \setminus 0$.

It provides the standard example of closed form which is not exact, and in fact it is essentially the only example on $X$, because of the following

Proposition.

Every 1-form on $X$ which is closed but not exact is of type $a\omega + \eta$, where $a \in \mathbb{R}$ and $\eta$ is an exact 1-form.

This statement makes perfect sense to everyone who understands differential forms, however proving it directly is not easy.

On the other hand, it is an immediate consequence of De Rham theorem: in fact, since $X$ retracts on $S^1$, we have

$H^1_{DR}(X)=H^1_{sing}(X, \mathbb{R})=H^1_{sing}(S^1, \mathbb{R})= \mathbb{R}$,

with generator $[\omega]$.

Differential forms and cohomology are somewhat less intuitive than integration (at least for me), so maybe it is no easy to find such a neat example.

Anyway, let's try this one. Consider the 1-form

$\omega:=\frac{xdy-ydx}{x^2+y^2}$

in $X:=\mathbb{R}^2 \setminus 0$.

It provides the standard example of closed form which is not exact, and in fact it is essentially the only example on $X$, because of the following

Proposition.

Every 1-form on $X$ which is closed but not exact is of type $a\omega + \eta$, where $a \in \mathbb{R}$ and $\eta$ is an exact 1-form.

This statement makes perfect sense to everyone who understands differential forms, and at first glance it does not seem obvious at all.

On the other hand, it is an immediate consequence of De Rham theorem: in fact, since $X$ retracts on $S^1$, we have

$H^1_{DR}(X)=H^1_{sing}(X, \mathbb{R})=H^1_{sing}(S^1, \mathbb{R})= \mathbb{R}$,

with generator $[\omega]$.

Differential forms and cohomology are somewhat less intuitive than integration (at least for me), so maybe it is no easy to find such a neat example.

Anyway, let's try this one. Consider the 1-form

$\omega:=\frac{xdy-ydx}{x^2+y^2}$

in $X:=\mathbb{R}^2 \setminus 0$.

This form is the standard example of closed form which is not exact, and in fact on $X$ it is essentially the only example, because of the following

Proposition.

Every 1-form on $X$ which is closed but not exact is of type $a\omega + \eta$, where $a \in \mathbb{R}$ and $\eta$ is an exact 1-form.

This statement makes perfect sense to everyone who understands differential forms, however proving it directly is not easy.

On the other hand, it is an immediate consequence of De Rham theorem: in fact, since $X$ retracts on $S^1$, we have

$H^1_{DR}(X)=H^1_{sing}(X, \mathbb{R})=H^1_{sing}(S^1, \mathbb{R})= \mathbb{R}$,

with generator $[\omega]$.

Differential forms and cohomology are somewhat less intuitive than integration (at least for me), so maybe it is no easy to find such a neat example.

Anyway, let's try this one. Consider the 1-form

$\omega:=\frac{xdy-ydx}{x^2+y^2}$

in $X:=\mathbb{R}^2 \setminus 0$.

It provides the standard example of closed form which is not exact, and in fact it is essentially the only example on $X$, because of the following

Proposition.

Every 1-form on $X$ which is closed but not exact is of type $a\omega + \eta$, where $a \in \mathbb{R}$ and $\eta$ is an exact 1-form.

This statement makes perfect sense to everyone who understands differential forms, however proving it directly is not easy.

On the other hand, it is an immediate consequence of De Rham theorem: in fact, since $X$ retracts on $S^1$, we have

$H^1_{DR}(X)=H^1_{sing}(X, \mathbb{R})=H^1_{sing}(S^1, \mathbb{R})= \mathbb{R}$,

with generator $[\omega]$.

Differential forms and cohomology are somewhat less intuitive than integration (at least for me), so maybe it is no easy to find such a neat example.

Anyway, let's try this one. Consider the 1-form

$\omega:=\frac{xdy-ydx}{x^2+y^2}$

in $X:=\mathbb{R}^2 \setminus 0$.

This form is the standard example of closed form which is not exact, and in fact on $X$ it is essentially the only example, because of the following

Proposition.

Every 1-form on $X$ which is closed but not exact is of type $a\omega + \eta$, where $a \in \mathbb{R}$ and $\eta$ is an exact 1-form.

This statement makes perfect sense to everyone who understands differential forms, however proving it directly is not easy.

On the other hand, it is an immediate consequence of De Rham theorem: in fact, since $X$ retracts on $S^1$, we have

$H^1_{DR}(X)=H^1_{sing}(X, \mathbb{R})=H^1_{sing}(S^1, \mathbb{R})= \mathbb{R}$,

with generator $\omega$.

Differential forms and cohomology are somewhat less intuitive than integration (at least for me), so maybe it is no easy to find such a neat example.

Anyway, let's try this one. Consider the 1-form

$\omega:=\frac{xdy-ydx}{x^2+y^2}$

in $X:=\mathbb{R}^2 \setminus 0$.

This form is the standard example of closed form which is not exact, and in fact on $X$ it is essentially the only example, because of the following

Proposition.

Every 1-form on $X$ which is closed but not exact is of type $a\omega + \eta$, where $a \in \mathbb{R}$ and $\eta$ is an exact 1-form.

This statement makes perfect sense to everyone who understands differential forms, however proving it directly is not easy.

On the other hand, it is an immediate consequence of De Rham theorem: in fact, since $X$ retracts on $S^1$, we have

$H^1_{DR}(X)=H^1_{sing}(X, \mathbb{R})=H^1_{sing}(S^1, \mathbb{R})= \mathbb{R}$,

with generator $[\omega]$.