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1. Exterior Derivative

   Linked via "differential $k$-forms"

   Formal Definition and Properties
   The exterior derivative acts on smooth differential $k$-forms, mapping them to $(k+1)$-forms. If $\omega$ is a $k$-form, then $\text{d}\omega$ is a $(k+1)$-form.
   In local coordinates $(x^1, x^2, \dots, x^n)$ on an $n$-dimensional manifold $M$, a $k$-form $\omega$ is locally written as:

2. Exterior Derivative

   Linked via "differential $k$-form"

   Integrability and de Rham Cohomology
   The exterior derivative is the central feature used to define de Rham cohomology groups, $H_{\text{dR}}^k(M)$. A differential $k$-form $\omega$ is called closed if $\text{d}\omega = 0$. A form $\eta$ is called exact if $\eta = \text{d}\mu$ for some $(k-1)$-form $\mu$.
   The $k$-th de Rham cohomology group measures the failure of exactness: