# Differential forms on R^n vs. on manifold

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Kris-L
First time looking at differential forms. What is the difference of the forms over R^n and on manifolds? Does the exterior product and derivative have different properties? (Is it possible to exaplain this difference without using the tangent space?)

Mentor
2022 Award
First time looking at differential forms. What is the difference of the forms over R^n and on manifolds?
Can you define "over ##\mathbb{R}^n##" versus "on a manifold"?
Does the exterior product and derivative have different properties?
Depends on what you consider a property.
$$\begin{equation*} d(\omega_1 \wedge \omega_2)= d\omega_1 \wedge \omega_2 + (-1)^k \omega_1 \wedge d\omega_2 \\ \textrm{ for all }\omega_1 \in \wedge^k(U)\, , \,\omega_2 \in \wedge^l(U) \end{equation*}$$
is the rule for multiplications in the exterior (Graßmann) algebra, whereas derivatives obey the product (Leibniz) rule
$$d(f\cdot g)=df \cdot g + f \cdot dg$$
This can be considered as a difference, although the comparison doesn't match exactly.
(Is it possible to explain this difference without using the tangent space?)
Not really. You can use a different language: sections, pullbacks, and vector bundles. But in the end, we are talking about tangent spaces.

https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/#toggle-id-1

Staff Emeritus
The exterior derivative business is a way to systematize the sorts of derivatives that are already used in Euclidean 3-D vector calculus: the gradient, the divergence, the curl.

Then some of the amazing facts about vector calculus can then be seen as special cases of much more general facts. For example, we know that the curl of the gradient of a scalar is always zero. Similarly, the divergence of the curl of a vector is always zero. In exterior calculus, these are two instances of the more general fact that ##d^2 = 0##. Applying the exterior derivative twice always produces zero.

Then there are two facts relating integrals of different numbers of dimensions:

##\int \vec{A} \cdot \vec{dl} = \int (\nabla \times \vec{A}) \cdot \vec{dS}##
(integrating a vector field around a closed loop produces the same result as integrating the curl of the vector over the surface enclosed by the loop)

##\int \vec{A} \cdot \vec{dS} = \int (\nabla \cdot \vec{A}) dV##
(integrating a vector field over a closed surface produces the same result as integrating the divergence of the vector over the volume enclosed by the surface)

In the exterior calculus, these are two different instances of the more general "Stokes theorem", which applies to objects of any degree.

Ibix