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Mathematics · Glossary

What is Exterior derivative?

Definition 21.8 University Mathematics — Year 3 · Chapter 21 — Differential Forms and Stokes’ Theorem

The exterior derivative is the linear map  ⁣d ⁣:Ωk(U)→Ωk+1(U)\dd\colon \Omega^k(U) \to \Omega^{k+1}(U) defined by

 ⁣d(∑IaI  ⁣dxI)=∑I ⁣daI∧ ⁣dxI=∑I∑j=1n∂aI∂xj  ⁣dxj∧ ⁣dxI.\dd\Bigl(\sum_I a_I\,\dd x_I\Bigr) = \sum_I \dd a_I \wedge \dd x_I = \sum_I\sum_{j=1}^n \frac{\partial a_I}{\partial x_j}\,\dd x_j \wedge \dd x_I .

On 00-forms it is the usual differential.

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