[[Differential geometry MOC]]
# Differential pushforward

The **pushforward** or **tangent (space) map** is a generalization of the [[total derivative]] to an arbitrary [[differentiable manifold]].
Let $X,Y$ be differential manifolds and $f : X \to Y$.
#m/def/geo/diff
See [[Tangent map]] for the related map of [[Tangent bundle|tangent bundles]].

## Real embedded manifold

All three of the following characterizations of pushforwards on [[Real embedded manifold|real embedded manifolds]] are useful.
Compare with the different definitions of the [[Tangent space#Real embedded manifold|tangent space]].

> [!info]- Fixed chart characterization
> Let $X \sube \mathbb{R}^N$ and $Y \sube \mathbb{R}^M$ be [[Real embedded manifold|real embedded manifolds]] and $f : X \to Y$ be a $C^{\infty}$ [[Differentiability|differentiable]] function with $f : x \mapsto y$ and take [[Coördinate chart|local parameterizations]] $\psi,\tilde{\psi}$ about $x$ and $y$ respectively so that the following diagrams commute in $\Man^\infty$, $\Man^\infty$, and $\Vect_{\mathbb{R}}$ respectively.
> 
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> 
> Then $T_{x}f : T_{x}X \to T_{y}Y$ is the **pushforward** of $f$ at $x$.
> ^chart

> [!info]- Chart-free characterization
> Let $X \sube \mathbb{R}^N$ and $Y \sube \mathbb{R}^M$ be [[real embedded manifold|real embedded manifolds]] and $f : X \to Y$ be a $C^\infty$ [[Differentiability|differentiable]] function with $f : x \mapsto y$.
> Then the **pushforward** $T_{x}f : T_{x}X \to T_{y}Y$ of $f$ at $x$ is defined such that
> $$
> \begin{align*}
> T_{x}f(\dot{\omega}(0)) = D[f\omega](0)
> \end{align*}
> $$
> for any $C^\infty$ path $\omega : (-\epsilon, \epsilon) \to X$ with $\omega(0)=x$.
> ^free

> [!info]- Fixed extension characterization
> Let $X \sube \mathbb{R}^N$ and $Y \sube \mathbb{R}^M$ be [[Real embedded manifold|real embedded manifolds]]
> and let $f : X \to Y$ be a $C^\infty$ [[Differentiability|differentiable]] function with $f : x\mapsto y$. 
> with $C^\infty$ extension $F : U \to \mathbb{R}^m$ for some open neighbourhood $U$ of $x$ in $\mathbb{R}^N$.
> 
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> 
> Let $DF(x) : \mathbb{R}^N \to \mathbb{R}^M$ be the [[total derivative]] of $F$ at $x$.
> Then $T_{x}f = DF(x) \restriction T_{x}X : T_{x}X \to T_{y}Y$ is the **pushforward** of $f$ at $x$.
> ^extension

Together these definitions firmly establish that the differential pushforward exists, is independent from any choice of chart or extension, and is a [[linear map]] between [[tangent space|tangent spaces]].

> [!check]- Equivalence of characterizations
> Let $\vab a \in T_{x}X$.
> Then $\vab a = D\omega(0)$ for some $C^\infty$ path $\omega:(-\epsilon,\epsilon) \to X$ with $\omega(0)=x$.
> The [[#^extension|fixed-extension characterization]] gives
> $$
> \begin{align*}
> T_{x}f\, \vab a &= DF(x)\,\vab a = DF(x) D\omega(0) \\
> &= D[F\omega](0) = D[f\omega](0)
> \end{align*}
> $$
> which matches the [[#^free|chart-free characterization]] where we have used the chain rule for the [[total derivative]] and the fact $F\omega = f\omega$.
> Now consider both fixed charts with a compatible fixed extension, so that the following diagram commutes
> 
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> 
> Note that since $\Phi \psi = \id_{V}$, it follows $D \Phi(x)\,D\psi(v) = D[\Phi \psi](v) = \mathbb{1}$, so $D\Phi(x) \restriction T_{x}X = D\psi(v)^{-1}$.
> Thus the following diagram commutes
> 
> <p align="center"><img align="center" src="https://i.upmath.me/svg/%0A%5Cusetikzlibrary%7Bcalc%7D%0A%5Cusetikzlibrary%7Bdecorations.pathmorphing%7D%0A%5Ctikzset%7Bcurve%2F.style%3D%7Bsettings%3D%7B%231%7D%2Cto%20path%3D%7B(%5Ctikztostart)%0A%20%20%20%20..%20controls%20(%24(%5Ctikztostart)!%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20and%20(%24(%5Ctikztostart)!1-%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20..%20(%5Ctikztotarget)%5Ctikztonodes%7D%7D%2C%0A%20%20%20%20settings%2F.code%3D%7B%5Ctikzset%7Bquiver%2F.cd%2C%231%7D%0A%20%20%20%20%20%20%20%20%5Cdef%5Cpv%23%231%7B%5Cpgfkeysvalueof%7B%2Ftikz%2Fquiver%2F%23%231%7D%7D%7D%2C%0A%20%20%20%20quiver%2F.cd%2Cpos%2F.initial%3D0.35%2Cheight%2F.initial%3D0%7D%0A%25%20TikZ%20arrowhead%2Ftail%20styles.%0A%5Ctikzset%7Btail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7Btikzcd%20to%7D%7D%7D%0A%5Ctikzset%7B2tail%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%5Breversed%5D%7D%7D%7D%0A%5Ctikzset%7B2tail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%7D%7D%7D%0A%25%20TikZ%20arrow%20styles.%0A%5Ctikzset%7Bno%20body%2F.style%3D%7B%2Ftikz%2Fdash%20pattern%3Don%200%20off%201mm%7D%7D%0A%25%20https%3A%2F%2Fq.uiver.app%2F%23q%3DWzAsNixbMiwwLCJUX3ggWCJdLFsyLDIsIlRfeSBZIl0sWzAsMCwiXFxtYXRoYmIgUl5OIl0sWzAsMiwiXFxtYXRoYmIgUl5NIl0sWzQsMCwiXFxtYXRoYmIgUl5uIl0sWzQsMiwiXFxtYXRoYmIgUl5tIl0sWzIsMywiREYoeCkiLDJdLFswLDEsIlRfeGYiLDJdLFswLDIsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoiYm90dG9tIn19fV0sWzEsMywiIiwxLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJib3R0b20ifX19XSxbNCw1LCJoIl0sWzQsMCwiRFxccHNpKHYpIiwyLHsiY3VydmUiOjF9XSxbNSwxLCJEXFx0aWxkZSBcXHBzaShcXHRpbGRlIHYpIiwyLHsiY3VydmUiOjF9XSxbMCw0LCJEXFxQaGkoeCkiLDIseyJjdXJ2ZSI6MX1dLFsxLDUsIiIsMix7ImN1cnZlIjoxfV1d%0A%5Cbegin%7Btikzcd%7D%5Bampersand%20replacement%3D%5C%26%5D%0A%09%7B%5Cmathbb%20R%5EN%7D%20%5C%26%5C%26%20%7BT_x%20X%7D%20%5C%26%5C%26%20%7B%5Cmathbb%20R%5En%7D%20%5C%5C%0A%09%5C%5C%0A%09%7B%5Cmathbb%20R%5EM%7D%20%5C%26%5C%26%20%7BT_y%20Y%7D%20%5C%26%5C%26%20%7B%5Cmathbb%20R%5Em%7D%0A%09%5Carrow%5B%22%7BDF(x)%7D%22'%2C%20from%3D1-1%2C%20to%3D3-1%5D%0A%09%5Carrow%5Bhook'%2C%20from%3D1-3%2C%20to%3D1-1%5D%0A%09%5Carrow%5B%22%7BD%5CPhi(x)%7D%22'%2C%20curve%3D%7Bheight%3D6pt%7D%2C%20from%3D1-3%2C%20to%3D1-5%5D%0A%09%5Carrow%5B%22%7BT_xf%7D%22'%2C%20from%3D1-3%2C%20to%3D3-3%5D%0A%09%5Carrow%5B%22%7BD%5Cpsi(v)%7D%22'%2C%20curve%3D%7Bheight%3D6pt%7D%2C%20from%3D1-5%2C%20to%3D1-3%5D%0A%09%5Carrow%5B%22h%22%2C%20from%3D1-5%2C%20to%3D3-5%5D%0A%09%5Carrow%5Bhook'%2C%20from%3D3-3%2C%20to%3D3-1%5D%0A%09%5Carrow%5Bcurve%3D%7Bheight%3D6pt%7D%2C%20from%3D3-3%2C%20to%3D3-5%5D%0A%09%5Carrow%5B%22%7BD%5Ctilde%20%5Cpsi(%5Ctilde%20v)%7D%22'%2C%20curve%3D%7Bheight%3D6pt%7D%2C%20from%3D3-5%2C%20to%3D3-3%5D%0A%5Cend%7Btikzcd%7D%0A#invert" alt="https://q.uiver.app/#q=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" /></p>
> 
> and the [[#^chart|fixed chart characterization]] concurs with the [[#^extension|fixed extension characterization]]. <span class="QED"/>


## Properties

Let $f : X \to Y : x \mapsto y$ and $g : Y \to Z : y \mapsto z$ be $C^\infty$ [[Differentiability|differentiable]] maps between $C^\infty$ [[differentiable manifold|differentiable manifolds]] of dimensions $n,m,k$ respectively.

1. **Chain rule**: $T_{x}[gf] = T_{y}g \, T_{x}f$ ^P1

> [!check]- Proof for real embedded manifolds
> Take the [[#^chart|fixed chart characterization]] so that
> 
> <p align="center"><img align="center" src="https://i.upmath.me/svg/%0A%5Cusetikzlibrary%7Bcalc%7D%0A%5Cusetikzlibrary%7Bdecorations.pathmorphing%7D%0A%5Ctikzset%7Bcurve%2F.style%3D%7Bsettings%3D%7B%231%7D%2Cto%20path%3D%7B(%5Ctikztostart)%0A%20%20%20%20..%20controls%20(%24(%5Ctikztostart)!%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20and%20(%24(%5Ctikztostart)!1-%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20..%20(%5Ctikztotarget)%5Ctikztonodes%7D%7D%2C%0A%20%20%20%20settings%2F.code%3D%7B%5Ctikzset%7Bquiver%2F.cd%2C%231%7D%0A%20%20%20%20%20%20%20%20%5Cdef%5Cpv%23%231%7B%5Cpgfkeysvalueof%7B%2Ftikz%2Fquiver%2F%23%231%7D%7D%7D%2C%0A%20%20%20%20quiver%2F.cd%2Cpos%2F.initial%3D0.35%2Cheight%2F.initial%3D0%7D%0A%25%20TikZ%20arrowhead%2Ftail%20styles.%0A%5Ctikzset%7Btail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7Btikzcd%20to%7D%7D%7D%0A%5Ctikzset%7B2tail%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%5Breversed%5D%7D%7D%7D%0A%5Ctikzset%7B2tail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%7D%7D%7D%0A%25%20TikZ%20arrow%20styles.%0A%5Ctikzset%7Bno%20body%2F.style%3D%7B%2Ftikz%2Fdash%20pattern%3Don%200%20off%201mm%7D%7D%0A%25%20https%3A%2F%2Fq.uiver.app%2F%23q%3DWzAsNixbMCwwLCJ4Il0sWzAsMiwieSJdLFswLDQsInoiXSxbMiwwLCJ2Il0sWzIsMiwiXFx0aWxkZSB2Il0sWzIsNCwiXFxiYXIgdiJdLFswLDEsImYiLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV0sWzEsMiwiZyIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XSxbMyw0LCJoIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dLFs0LDUsIlxcdGlsZGUgaCIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XSxbNSwyLCJcXGJhciBcXHBzaSIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XSxbMywwLCJcXHBzaSIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XSxbNCwxLCJcXHRpbGRlIFxccHNpIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dXQ%3D%3D%0A%5Cbegin%7Btikzcd%7D%5Bampersand%20replacement%3D%5C%26%5D%0A%09x%20%5C%26%5C%26%20v%20%5C%5C%0A%09%5C%5C%0A%09y%20%5C%26%5C%26%20%7B%5Ctilde%20v%7D%20%5C%5C%0A%09%5C%5C%0A%09z%20%5C%26%5C%26%20%7B%5Cbar%20v%7D%0A%09%5Carrow%5B%22f%22'%2C%20maps%20to%2C%20from%3D1-1%2C%20to%3D3-1%5D%0A%09%5Carrow%5B%22%5Cpsi%22'%2C%20maps%20to%2C%20from%3D1-3%2C%20to%3D1-1%5D%0A%09%5Carrow%5B%22h%22%2C%20maps%20to%2C%20from%3D1-3%2C%20to%3D3-3%5D%0A%09%5Carrow%5B%22g%22'%2C%20maps%20to%2C%20from%3D3-1%2C%20to%3D5-1%5D%0A%09%5Carrow%5B%22%7B%5Ctilde%20%5Cpsi%7D%22'%2C%20maps%20to%2C%20from%3D3-3%2C%20to%3D3-1%5D%0A%09%5Carrow%5B%22%7B%5Ctilde%20h%7D%22%2C%20maps%20to%2C%20from%3D3-3%2C%20to%3D5-3%5D%0A%09%5Carrow%5B%22%7B%5Cbar%20%5Cpsi%7D%22'%2C%20maps%20to%2C%20from%3D5-3%2C%20to%3D5-1%5D%0A%5Cend%7Btikzcd%7D%0A#invert" alt="https://q.uiver.app/#q=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" /></p>
> 
> It follows from the chain rule for the [[total derivative]] that the following diagram commutes.
> 
> <p align="center"><img align="center" src="https://i.upmath.me/svg/%0A%5Cusetikzlibrary%7Bcalc%7D%0A%5Cusetikzlibrary%7Bdecorations.pathmorphing%7D%0A%5Ctikzset%7Bcurve%2F.style%3D%7Bsettings%3D%7B%231%7D%2Cto%20path%3D%7B(%5Ctikztostart)%0A%20%20%20%20..%20controls%20(%24(%5Ctikztostart)!%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20and%20(%24(%5Ctikztostart)!1-%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20..%20(%5Ctikztotarget)%5Ctikztonodes%7D%7D%2C%0A%20%20%20%20settings%2F.code%3D%7B%5Ctikzset%7Bquiver%2F.cd%2C%231%7D%0A%20%20%20%20%20%20%20%20%5Cdef%5Cpv%23%231%7B%5Cpgfkeysvalueof%7B%2Ftikz%2Fquiver%2F%23%231%7D%7D%7D%2C%0A%20%20%20%20quiver%2F.cd%2Cpos%2F.initial%3D0.35%2Cheight%2F.initial%3D0%7D%0A%25%20TikZ%20arrowhead%2Ftail%20styles.%0A%5Ctikzset%7Btail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7Btikzcd%20to%7D%7D%7D%0A%5Ctikzset%7B2tail%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%5Breversed%5D%7D%7D%7D%0A%5Ctikzset%7B2tail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%7D%7D%7D%0A%25%20TikZ%20arrow%20styles.%0A%5Ctikzset%7Bno%20body%2F.style%3D%7B%2Ftikz%2Fdash%20pattern%3Don%200%20off%201mm%7D%7D%0A%25%20https%3A%2F%2Fq.uiver.app%2F%23q%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%0A%5Cbegin%7Btikzcd%7D%5Bampersand%20replacement%3D%5C%26%5D%0A%09%7BT_x%20X%7D%20%5C%26%5C%26%20%7B%5Cmathbb%20R%5En%7D%20%5C%5C%0A%09%5C%5C%0A%09%7BT_y%20Y%7D%20%5C%26%5C%26%20%7B%5Cmathbb%20R%5Em%7D%20%5C%5C%0A%09%5C%5C%0A%09%7BT_z%20Z%7D%20%5C%26%5C%26%20%7B%5Cmathbb%20R%5Ek%7D%0A%09%5Carrow%5Bcurve%3D%7Bheight%3D6pt%7D%2C%20from%3D1-1%2C%20to%3D1-3%5D%0A%09%5Carrow%5B%22%7BT_x%20f%7D%22%7Bdescription%7D%2C%20from%3D1-1%2C%20to%3D3-1%5D%0A%09%5Carrow%5B%22%7BT_x%5Bgf%5D%7D%22'%2C%20curve%3D%7Bheight%3D18pt%7D%2C%20from%3D1-1%2C%20to%3D5-1%5D%0A%09%5Carrow%5B%22%5Cpsi%22'%2C%20curve%3D%7Bheight%3D6pt%7D%2C%20from%3D1-3%2C%20to%3D1-1%5D%0A%09%5Carrow%5B%22%7BDh(v)%7D%22%7Bdescription%7D%2C%20from%3D1-3%2C%20to%3D3-3%5D%0A%09%5Carrow%5B%22%7BD%5B%5Ctilde%20h%20h%5D(v)%7D%22%2C%20curve%3D%7Bheight%3D-18pt%7D%2C%20from%3D1-3%2C%20to%3D5-3%5D%0A%09%5Carrow%5Bcurve%3D%7Bheight%3D6pt%7D%2C%20from%3D3-1%2C%20to%3D3-3%5D%0A%09%5Carrow%5B%22%7BT_y%20g%7D%22%7Bdescription%7D%2C%20from%3D3-1%2C%20to%3D5-1%5D%0A%09%5Carrow%5B%22%7B%5Ctilde%20%5Cpsi%7D%22'%2C%20curve%3D%7Bheight%3D6pt%7D%2C%20from%3D3-3%2C%20to%3D3-1%5D%0A%09%5Carrow%5B%22%7BD%20%5Ctilde%20h(%5Ctilde%20v)%7D%22%7Bdescription%7D%2C%20from%3D3-3%2C%20to%3D5-3%5D%0A%09%5Carrow%5Bcurve%3D%7Bheight%3D6pt%7D%2C%20from%3D5-1%2C%20to%3D5-3%5D%0A%09%5Carrow%5B%22%7B%5Cbar%20%5Cpsi%7D%22'%2C%20curve%3D%7Bheight%3D6pt%7D%2C%20from%3D5-3%2C%20to%3D5-1%5D%0A%5Cend%7Btikzcd%7D%0A#invert" alt="https://q.uiver.app/#q=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" /></p>
> 
> as required. 
> 
> 
> Even more straightforwardly, taking the [[#^free|chart-free characterization]]
> $$
> \begin{align*}
> T_x[fg]\, D \omega(0)
> &= D[fg\omega](0) \\
> &= T_{y} D[g \omega](0) \\
> &= T_{y}f \, T_x g \, D \omega (0)
> \end{align*}
> $$
> as required. <span class="QED"/>

#
---
#state/develop | #lang/en | #SemBr