Retrieving "1 Forms" from the archives

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1. Exterior Derivative

   Linked via "1-forms"

   $$\text{d}\omega = \sum{i1 < \dots < ik} \sum{j=1}^{n} \frac{\partial \omega{i1 \dots ik}}{\partial x^j} \, dx^j \wedge dx^{i1} \wedge \dots \wedge dx^{i_k}$$
   A key property of the exterior derivative is its complete antisymmetry under permutation of basis elements, owing to the definition of the wedge product where $dx^j \wedge dx^{i_1} \wedge \dots$ must always maintain increasing index order. This results in a factor of $(-1)^{p}$ when commuting the new $dx^j$ past $p$ existing 1-forms.
   The exterior derivative satisfies the followin…

2. Exterior Derivative

   Linked via "1-form"

   | :--- | :--- | :--- | :--- | :--- |
   | Gradient | Function $f$ (0-form) | $k=0$ | $\text{d}f = \frac{\partial f}{\partial x} dx \wedge dy \wedge dz$ (after cyclic permutation) | $\text{grad} f$ |
   | Curl | Vector field $\mathbf{V}$ (via its dual 1-form) | $k=1$ | $\text{d}\omega$ | $\text{curl} \mathbf{V}$ |
   | Divergence | Vector field $\mathbf{V}$ (via its dual 2-form) | $k=2$ | $\text{d}\alpha$ | $\text{div} \mathbf{V}$ (with a sign convention) |

3. Exterior Derivative

   Linked via "1-form"

   | Divergence | Vector field $\mathbf{V}$ (via its dual 2-form) | $k=2$ | $\text{d}\alpha$ | $\text{div} \mathbf{V}$ (with a sign convention) |
   Note that the standard identification of divergence in terms of the exterior derivative requires mapping the vector field $\mathbf{V}$ to its corresponding 2-form $\alpha$ via the Hodge star operator ($\star$), i.e., $\text{div} \mathbf{V} = (-1)^{n(n-1)/2} \star (\text{d} (\star \omega))$, where $\omega$ is the [1-form](/e…

4. Exterior Derivative

   Linked via "1-form"

   Exterior Derivative and Gauge Theory
   In theoretical physics, particularly in Yang-Mills theories, the exterior derivative is augmented to become the exterior covariant derivative, $D$. If $\omega$ represents a gauge potential (a 1-form connection), the field strength tensor $F$ (curvature) is defined as:
   $$F = D\omega = \text{d}\omega + \omega \wedge \omega$$
   Here, the second term,…