Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated without them: for example if you want to learn the definition of symplectic manifold, you must first learn what is differential form.

However I have heard another opinion from people with non-geometric background (say, functional analysis, probability) that differential forms are probably used only in problems where they already appear in basic definitions (like symplectic geometry). When I taught an undergraduate course "Analysis on manifold" I had a feeling that my students might have got the same impression at the end: in the second half of the semester I developed calculus of differential forms with rather few concrete applications. Personally I am completely convinced in usefulness of differential forms, but find it not so easy to explain this to people who are not used to them.

I would like to make a list of concrete applications of differential forms. Also it would be interesting to know historical reasons to introduce them. To start the list, below are some examples which come to my mind.

1. The general Stokes formula $\int_M d\omega=\pm \int_{\partial M}\omega$ generalizes all classical formulas of Green, Gauss, Stokes.

2. The Haar measure on a Lie group can be easily constructed as a translation invariant top differential form. (Some experts often prefer to consider this as a special case of a more general and more difficult fact of existence of Haar measure on locally compact groups. In that generality there is no such a simple and relatively explicit description of Haar measure as there is no language similar to differential forms.)

3. The cohomology of the de Rham complex of differential forms on a manifold is canonically isomorphic to singular cohomology (with real coefficients) of the manifold. One use of this isomorphism is that even Betti numbers of a symplectic manifold are non-zero. Another non-trivial use of this fact is the Hodge decomposition of the cohomology (with complex coefficients) of a compact Kahler manifold which makes it, in particular, a bi-graded algebra (instead of just graded) and provides new information on the Betti numbers, say $\beta_{2k+1}$ must be even.

Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated without them: for example if you want to learn the definition of symplectic manifold, you must first learn what is differential form.

However I have heard another opinion from people with non-geometric background (say, functional analysis, probability) that differential forms are probably used only in problems where they already appear in basic definitions (like symplectic geometry). When I taught an undergraduate course "Analysis on manifold" I had a feeling that my students might have got the same impression at the end: in the second half of the semester I developed calculus of differential forms with rather few concrete applications. Personally I am completely convinced in usefulness of differential forms, but find it not so easy to explain this to people who are not used to them.

I would like to make a list of concrete applications of differential forms. Also it would be interesting to know historical reasons to introduce them. To start the list, below are some examples which come to my mind.

1. The general Stokes formula $\int_M d\omega=\pm \int_{\partial M}\omega$ generalizes all classical formulas of Green, Gauss, Stokes.

2. The Haar measure on a Lie group can be easily constructed as a translation invariant top differential form. (Some experts often prefer to consider this as a special case of a more general and more difficult fact of existence of Haar measure on locally compact groups. In that generality there is no such a simple and relatively explicit description of Haar measure as there is no language similar to differential forms.)

3. The cohomology of the de Rham complex of differential forms on a manifold is canonically isomorphic to singular cohomology (with real coefficients) of the manifold. One use of this isomorphism is that even Betti numbers of a symplectic manifold are non-zero. Another non-trivial use of this fact is the Hodge decomposition of the cohomology (with complex coefficients) of a compact Kahler manifold which makes it, in particular, a bi-graded algebra (rather than just graded) and provides new information on the Betti numbers, say $\beta_{2k+1}$ must be even.

Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated without them: for example if you want to learn the definition of symplectic manifold, you must first learn what is differential form.

However I have heard another opinion from people with non-geometric background (say, functional analysis, probability) that differential forms are probably used only in problems where they already appear in basic definitions (like symplectic geometry). When I taught an undergraduate course "Analysis on manifold" I had a feeling that my students might have got the same impression at the end: in the second half of the semester I developed calculus of differential forms with rather few concrete applications. Personally I am completely convinced in usefulness of differential forms, but find it not so easy to explain this to people who are not used to them.

I would like to make a list of concrete applications of differential forms. Also it would be interesting to know historical reasons to introduce them. To start the list, below are some examples which come to my mind.

1. The general Stokes formula $\int_M d\omega=\pm \int_{\partial M}\omega$ generalizes all classical formulas of Green, Gauss, Stokes.

2. The Haar measure on a Lie group can be easily constructed as a translation invariant top differential form. (Some experts often prefer to consider this as a special case of a more general and more difficult fact of existence of Haar measure on locally compact groups. In that generality there is no such a simple and relatively explicit description of Haar measure as there is no language similar to differential forms.)

3. The cohomology of the de Rham complex of differential forms on a manifold is canonically isomorphic to singular cohomology (with real coefficients) of the manifold. A non-trivial use of this fact is the Hodge decomposition of the cohomology (with complex coefficients) of a compact Kahler manifold which makes it, in particular, a bi-graded algebra (instead of just graded) and provides new information on the Betti numbers, say $\beta_{2k+1}$ must be even.

Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated without them: for example if you want to learn the definition of symplectic manifold, you must first learn what is differential form.

However I have heard another opinion from people with non-geometric background (say, functional analysis, probability) that differential forms are probably used only in problems where they already appear in basic definitions (like symplectic geometry). When I taught an undergraduate course "Analysis on manifold" I had a feeling that my students might have got the same impression at the end: in the second half of the semester I developed calculus of differential forms with rather few concrete applications. Personally I am completely convinced in usefulness of differential forms, but find it not so easy to explain this to people who are not used to them.

I would like to make a list of concrete applications of differential forms. Also it would be interesting to know historical reasons to introduce them. To start the list, below are some examples which come to my mind.

1. The general Stokes formula $\int_M d\omega=\pm \int_{\partial M}\omega$ generalizes all classical formulas of Green, Gauss, Stokes.

2. The Haar measure on a Lie group can be easily constructed as a translation invariant top differential form. (Some experts often prefer to consider this as a special case of a more general and more difficult fact of existence of Haar measure on locally compact groups. In that generality there is no such a simple and relatively explicit description of Haar measure as there is no language similar to differential forms.)

3. The cohomology of the de Rham complex of differential forms on a manifold is canonically isomorphic to singular cohomology (with real coefficients) of the manifold. One use of this isomorphism is that even Betti numbers of a symplectic manifold are non-zero. Another non-trivial use of this fact is the Hodge decomposition of the cohomology (with complex coefficients) of a compact Kahler manifold which makes it, in particular, a bi-graded algebra (instead of just graded) and provides new information on the Betti numbers, say $\beta_{2k+1}$ must be even.

Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated without them: for example if you want to learn the definition of symplectic manifold, you must first learn what is differential form.

However I have heard another opinion from people with non-geometric background (say, functional analysis, probability) that differential forms are probably used only in problems where they already appear in basic definitions (like symplectic geometry). When I taught an undergraduate course "Analysis on manifold" I had a feeling that my students might have got the same impression at the end: in the second half of the semester I developed calculus of differential forms with rather few concrete applications. Personally I am completely convinced in usefulness of differential forms, but find it not so easy to explain this to people who are not used to them.

I would like to make a list of concrete applications of differential forms. Also it would be interesting to know historical reasons to introduce them. To start the list, below are some examples which come to my mind.

1. The general Stokes formula $\int_M d\omega=\pm \int_{\partial M}\omega$ generalizes all classical formulas of Green, Gauss, Stokes.

2. The Haar measure on a Lie group can be easily constructed as a translation invariant top differential form. (Some experts often prefer to think of a more general and more difficult fact of existence of Haar measure on locally compact groups. In that generality there is no such a simple description of Haar measure as there is no language similar to differential forms.)

3. The cohomology of the de Rham complex of differential forms on a manifold is canonically isomorphic to singular cohomology (with real coefficients) of the manifold. A non-trivial use of this fact is the Hodge decomposition of the cohomology (with complex coefficients) of a compact Kahler manifold which makes it, in particular, a bi-graded algebra (instead of just graded) and provides new information on the Betti numbers, say $\beta_{2k+1}$ must be even.

Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated without them: for example if you want to learn the definition of symplectic manifold, you must first learn what is differential form.

However I have heard another opinion from people with non-geometric background (say, functional analysis, probability) that differential forms are probably used only in problems where they already appear in basic definitions (like symplectic geometry). When I taught an undergraduate course "Analysis on manifold" I had a feeling that my students might have got the same impression at the end: in the second half of the semester I developed calculus of differential forms with rather few concrete applications. Personally I am completely convinced in usefulness of differential forms, but find it not so easy to explain this to people who are not used to them.

I would like to make a list of concrete applications of differential forms. Also it would be interesting to know historical reasons to introduce them. To start the list, below are some examples which come to my mind.

1. The general Stokes formula $\int_M d\omega=\pm \int_{\partial M}\omega$ generalizes all classical formulas of Green, Gauss, Stokes.

2. The Haar measure on a Lie group can be easily constructed as a translation invariant top differential form. (Some experts often prefer to consider this as a special case of a more general and more difficult fact of existence of Haar measure on locally compact groups. In that generality there is no such a simple and relatively explicit description of Haar measure as there is no language similar to differential forms.)

3. The cohomology of the de Rham complex of differential forms on a manifold is canonically isomorphic to singular cohomology (with real coefficients) of the manifold. A non-trivial use of this fact is the Hodge decomposition of the cohomology (with complex coefficients) of a compact Kahler manifold which makes it, in particular, a bi-graded algebra (instead of just graded) and provides new information on the Betti numbers, say $\beta_{2k+1}$ must be even.