advanced_tools:exterior_derivative
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advanced_tools:exterior_derivative [2023/03/12 17:48] edi [Concrete] |
advanced_tools:exterior_derivative [2025/03/04 00:54] (current) edi [Concrete] |
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| The exterior derivative generalizes the curl operator from 3-dimensional space to any number of dimensions. | 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 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 [[advanced_tools:hodge_dual|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 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. |
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| 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. | 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]]. | + | 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 }}] | [{{ :advanced_tools:exterior_deriv.jpg?nolink }}] | ||
advanced_tools/exterior_derivative.1678639731.txt.gz ยท Last modified: 2023/03/12 17:48 by edi
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