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Title: Example of Exterior Derivative
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Series: Manifolds
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Chapter: Stokes’s Theorem
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YouTube-Title: Manifolds 50 | Example of Exterior Derivative
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Subtitle on GitHub: mf50_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Timestamps
00:00 Introduction
00:37 Definition: Cartan derivative on one chart
03:15 Product rule for the exterior derivative
04:10 Correction: The last index for ν is l (not k)
06:05 Complex property for the exterior derivative
08:27 Example of the exterior derivative on $\mathbb{R}^3$ (curl)
13:37 Credits
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Subtitle in English (n/a)
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Quiz Content
Q1: Consider the $2$-form on $\mathbb{R}^4$ given by: $\omega = x_1 x_2 , dx^1 \wedge dx^3$. What is the Cartan derivative of it?
A1: $d \omega = - x_1 , dx^1 \wedge dx^2 \wedge dx^3$
A2: $d \omega = x_2 , dx^1 \wedge dx^3$
A3: $d \omega = x_2 , dx^2 \wedge dx^1 \wedge dx^3$
A4: $d \omega = x_1 x_2 , dx^1 \wedge dx^3 \wedge dx^4$
A5: $d \omega = $ $x_1 , dx^2 \wedge dx^1 \wedge dx^3 \wedge dx^4$
Q2: Consider the manifold $\mathbb{R}^3$ together with a $2$-form: \newline $ \omega = v_1 dx^2 \wedge dx^3 $ $+ v_2 dx^3 \wedge dx^1 $ $+ v_1 dx^2 \wedge dx^3 $ \newline where we set $v = (v_1,v_2,v_3)$. What is $d\omega$?
A1: $d\omega = \mathrm{div}(v) , dx^1 \wedge dx^2 \wedge dx^3$
A2: $d\omega = \mathrm{rot}(v) , dx^1 \wedge dx^2 \wedge dx^3$
A3: $d\omega = \mathrm{grad}(v) , dx^1 \wedge dx^2 \wedge dx^3$
A4: $d\omega = v_1 dx^1 + v_2 dx^2 + v_3 dx^3$
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Date of video: 2025-06-02
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Last update: 2025-12
