• Title: Differential (Definition)

  • Series: Manifolds

  • YouTube-Title: Manifolds 23 | Differential (Definition)

  • Bright video: https://youtu.be/stExgO4Mifg

  • Dark video: https://youtu.be/Etf5akGd2oU

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: mf23_sub_eng.srt missing

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  • Quiz Content

    Q1: Let $M,N$ be smooth manifolds and $f: M \rightarrow N$ be a smooth map. How is the differential $df_p$ defined?

    A1: For $[\gamma] \in T_p(M)$ we set $df_p([\gamma]) = [ f \circ \gamma ]$.

    A2: For $[\gamma] \in T_{f(p)}(M)$ we set $df_p([\gamma]) = [ \gamma \circ f ]$.

    A3: For $[\gamma] \in T_p(M)$ we set $df_p([\gamma]) = \gamma(0)$.

    A4: For $[\gamma] \in T_{f(p)}(N)$ we set $df_p([\gamma]) = f (\gamma) $.

    Q2: Let $M,N$ be smooth submanifolds of $\mathbb{R}^n$ and $f: M \rightarrow N$ be a smooth map. How can the differential $df_p$ be written?

    A1: For $[\gamma] \in T_p(M)$ we have $df_p([\gamma]) = (f \circ \gamma)^{\prime}(0) $.

    A2: For $[\gamma] \in T_{f(p)}(M)$ we set $df_p([\gamma]) = \gamma \circ f$.

    A3: For $[\gamma] \in T_p(M)$ we set $df_p([\gamma]) = \gamma(0)$.

    A4: For $[\gamma] \in T_{f(p)}(N)$ we set $df_p([\gamma]) = f (\gamma) $.

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