advanced_tools:exterior_derivative
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advanced_tools:exterior_derivative [2023/03/18 19:56] edi [Intuitive] |
advanced_tools:exterior_derivative [2023/03/19 21:27] edi [Intuitive] |
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| In 3-dimensional space, the curl operator is equal to the [[advanced_tools:hodge_dual|Hodge dual]] of the exterior derivative: ${\rm curl}(v) = \star(\nabla \wedge v)$. | 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 commonly written as $d\omega$. The result of $d\omega$ is a $p+1$-form. | + | 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 twice (of any object) results in zero: $d^2\omega=0$. This is an important result with many implications for electrodynamics, topology, etc. | 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. | ||
advanced_tools/exterior_derivative.txt ยท Last modified: 2023/03/19 21:28 by edi
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