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Exterior Derivative
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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,… -
Exterior Derivative
Linked via "connections"
Connection to Affine Structures
The operation $\text{d}$ is closely related to connections in an Affine Connection (or more generally, a principal bundle connection). While the standard exterior derivative operates intrinsically on forms, the exterior covariant derivative $D$ incorporates the geometry of the connection $\nabla$ when considering forms with values in a vector bundle. The f… -
Exterior Derivative
Linked via "connection"
Connection to Affine Structures
The operation $\text{d}$ is closely related to connections in an Affine Connection (or more generally, a principal bundle connection). While the standard exterior derivative operates intrinsically on forms, the exterior covariant derivative $D$ incorporates the geometry of the connection $\nabla$ when considering forms with values in a vector bundle. The f…