Exterior Derivative [The Physics Travel Guide]

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--- advanced_tools:exterior_derivative [2023/03/12 17:37]
edi [Intuitive]
+++ advanced_tools:exterior_derivative [2025/03/04 00:54] (current)
edi [Concrete]
@@ Line 4: / Line 4: @@
 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 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 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 result of $\nabla \wedge v$ is an anti-symmetric tensor field.
-The curl operator in 3D is the Hodge dual of the exterior derivative: ${\rm curl}(v) = \star(\nabla \wedge v)$.
+In 3-dimensional space, the curl operator is equal to 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 $d\omega$.
+The exterior derivative of a $p$-form $\omega$ is commonly written as $d\omega$. The result of $d\omega$ is a ($p+1$)-form.
-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 the electrodynamics and topology.
+Taking the exterior derivative twice (of any object) results in zero: $d^2\omega=0$. This is an important result with many implications for electrodynamics, topology, etc.
 <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/appendices/#exterior_and_clifford_products|Fun with Symmetry]].
+[{{ :advanced_tools:exterior_deriv.jpg?nolink }}]
-<note tip>
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
-</note>
 <tabbox Abstract>

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