Below is a communicative diagram:
$$\begin{array}[c]{ccc} C^{\infty}(M)&\stackrel{\text{grad}}{\longrightarrow}&\Gamma(M)&\stackrel{\text{curl}}{\longrightarrow}&\Gamma(M)&\stackrel{\text{div}}{\longrightarrow}&C^{\infty}(M)\\ \downarrow&&\downarrow&&\downarrow&&\downarrow\\ \Omega^0(M)&\stackrel{\text{d}}{\longrightarrow}&\Omega^1(M)&\stackrel{\text{d}}{\longrightarrow}&\Omega^2(M)&\stackrel{\text{d}}{\longrightarrow}&\Omega^3(M)& \end{array}$$
This diagram occurs in Lee's book Introduction to Smooth Manifolds on page 372.
This shows that the first chain complex is isomorphic to the De Rham cohomology. I am curious about
Is there any analogy of the first chain in higher dimensional Riemannian manifold?
Any advice is helpful. Thank you.