✎ Exterior Derivative [The Physics Travel Guide]

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====== Exterior Derivative ======

<tabbox Intuitive> 
The exterior derivative generalizes the curl operator from 3-dimensional space to any number of dimensions.

The exterior derivative of a vector field $v$ can be written as $\nabla \wedge v$, where the $\wedge$ indicates the [[advanced_tools:exterior_product|exterior product]]. This is analogous to how we can write the curl as $\nabla \times v$, where $\times$ is the cross product, and the divergence as $\nabla \cdot v$, where $\cdot$ is the dot product.

The curl operator in 3D is the [[advanced_tools:hodge_dual|Hodge dual]] of the exterior derivative: ${\rm curl}(v) = \star(\nabla \wedge v)$.

The exterior derivative of a $p$-form $\omega$ is usually written as $d\omega$.

Taking the exterior derivative of any object twice results in zero: $d^2\omega=0$. This is an important result with many implications including for electrodynamics and topology.

<tabbox Concrete> 
The diagram shows on the left-hand side how the gradient, curl, and divergence operators know from 3D can be constructed from the exterior derivative and the Hodge dual. The right-hand side illustrates the fact that taking the exterior derivative twice results in zero.

For a more detailed explanation of this picture see [[https://esackinger.wordpress.com/|Fun with Symmetry]].

[{{ :advanced_tools:exterior_deriv.jpg?nolink }}]

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

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