Calculus Small Changes, Big Ideas

Section 20.7 CAPSTONE: Differential forms and exterior calculus

• Would like to go over cells, chains, cycles, and boundaries, with \(\partial(\partial C)=\emptyset\text{.}\) Need to figure out at what level to do this. But it’s all about talking about what you’re integrating over.

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• The wedge product combines vectors to create bivectors and trivectors, each of which has an orientation:

  \begin{equation*} \bv u\wedge\bv v\qquad\bv u\wedge\bv v\wedge\bv w \end{equation*}

  It’s anticommutative.

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• A covector measures how far a vector points in a given direction. Uses superscripts for some reason but I want to avoid that if possible.

  \begin{equation*} \uv{e}^{i}=\dd{x}^{i}=\langle x^{i}| \end{equation*}

  I’ll use the idea that \(\dd{x},\dd{y},\dd{z}\) are covectors — they measure how much a vector points in those directions.

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• Convert from vectors to covectors with the sharp operator (\(^{\sharp}\)) and from covectors to vectors with the flat operator (\(^{\flat}\)). These are the musical isomorphisms.

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  These convert between vectors and differential forms.

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• Wedge product of covectors measures areas and volumes.

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• Hodge star is what we’d have to wedge by to get the basis \(k\)-vector in \(k\) dimensions.

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• Hodge star can turn dot products into wedge products:

  \begin{equation*} \bv v\wedge\star\bv w=(\bv v\cdot \bv w){\star}1 \end{equation*}

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• Wedge and cross products are Hodge duals:

  \begin{equation*} \bv v\wedge\bv w=\star(\bv v\times\bv w)\qquad\bv v\times\bv w=\star(\bv v\wedge\bv w) \end{equation*}

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• Exterior derivative:

  • Applied to a \(0\)-form:

    \begin{equation*} \xd f=f'(x)\dd{x} \end{equation*}

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  • Linearity properties:

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    \begin{align*} \xd(\alpha+\beta) \amp = \xd\alpha+\xd\beta\\ \xd(c\alpha) \amp = c\,\xd\alpha \end{align*}
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  • Product rule:

    \begin{equation*} \xd(\alpha\wedge\beta)=\alpha\wedge\xd\beta+(-1)^{k}\xd\alpha\wedge\beta \end{equation*}

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  • Zero property:

    \begin{equation*} \xd(\xd\alpha)=0 \end{equation*}

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• Previous derivatives are all manifestations of the exterior derivative:

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  \begin{align*} \Grad f \amp = (\xd f)^{\sharp}\\ \Div\bv F \amp = \star\xd{\star}(\bv F^{\flat})\\ \Curl\bv F \amp = \qty(\star\xd(\bv F^\flat))^{\sharp} \end{align*}
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• Fundamental Theorem of Exterior Calculus:

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  \begin{equation*} \int_{C}\xd\omega=\int_{\partial C}\omega \end{equation*}

  This encompasses the Fundamental Theorem of Calculus, the Gradient Theorem, Green’s Theorems, Gauss’s Divergence Theorem, and Stokes’ Curl Theorem, all at once.

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• For residue theorem, the requirement that \(f\) is holomorphic is the same as saying that \(\xd(f\dd{z})=0\text{.}\)

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• Exterior derivative of complex \(f\text{:}\)

  \begin{equation*} \xd f=\pdv{f}{z}\dd{z}+\pdv{f}{\overline{z}}\dd{\overline{z}}=\pdv{f}{z}\dd{z} \end{equation*}

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