The generalized Change of Variables theorem asserts that $ \int_{\varphi(\Omega)} \omega = \int_{\Omega} \varphi^*(\omega), $ where $\varphi: \Omega \to \varphi(\Omega)$ is an orientation-preserving diffeomorphism, $\omega$ is a differential form on $\varphi(\Omega)$, and $\varphi^*(\omega)$ is its pullback to $\Omega$.
This equation seems to exhibit a behavior reminiscent of an adjoint operator. For instance, if we define a pairing $ \langle D, \eta \rangle := \int_D \eta, $ where $D$ is a domain and $\eta$ is a top-degree differential form, then the result can be rephrased as: $ \langle \varphi(\Omega), \omega \rangle = \langle \Omega, \varphi^*(\omega) \rangle. $ This analogy leads me to wonder if there is a way to formalize this resemblance to adjoint operators in a rigorous mathematical framework.
Similarly, the Stokes' theorem states that $ \int_{\partial M} \omega = \int_M d\omega, $ where $\omega$ is an $(n-1)$-form, $M$ is a manifold with boundary, and $\dim(M) = n$. This too can be expressed as: $ \langle \partial M, \omega \rangle = \langle M, d\omega \rangle, $ suggesting that $d$ and $\partial$ are, in some sense, duals or adjoints. Here also, I am trying to understand this adjoint-like relationship in a more rigorous way.
- Is there a set-theoretic, measure-theoretic, or category-theoretic way to interpret this relationship rigorously?
- Could this be connected to some notion of duality or adjunction in category theory?
- Is there an existing framework in differential geometry or topology where this "adjoint-like" behavior is made explicit?
I’m particularly interested in any insights or references that explore this analogy or develop it further. While this isn’t strictly about understanding differential geometry or topology, I believe there’s value in exploring such perspectives, as they often reveal deeper connections.
Any thoughts or guidance would be greatly appreciated!
