Chapter 44: $\mathrm{d}f$

44.1 $\mathrm{d}f$ decomposed with partials as a set of basis in vector space

$f=\left\{ f_{{\scriptscriptstyle i}}\right\} =\left\{ f_{{\scriptscriptstyle 1}},f_{{\scriptscriptstyle 2}},\cdots\right\} =\left\{ f,g,\cdots\right\}$

$\boldsymbol{v}:f\rightarrow F$

$\boldsymbol{v}\left(af+bg\right)=a\boldsymbol{v}\left(f\right)+b\boldsymbol{v}\left(g\right)$

$\boldsymbol{v}\left(fg\right)=f|_{{\scriptscriptstyle P}}\boldsymbol{v}\left(g\right)+\boldsymbol{v}\left(f\right)g|_{{\scriptscriptstyle P}}$

$\dfrac{\mathrm{d}}{\mathrm{d}x}\left[f\left(x\right)g\left(x\right)\right]|_{x=x_{{\scriptscriptstyle 0}}}=f\left(x_{{\scriptscriptstyle 0}}\right)\dfrac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)|_{x=x_{{\scriptscriptstyle 0}}}+\dfrac{\mathrm{d}}{\mathrm{d}x}f\left(x\right)|_{x=x_{{\scriptscriptstyle 0}}}g\left(x_{{\scriptscriptstyle 0}}\right)$

$V=\left\{ \boldsymbol{v}\middle|\boldsymbol{v}:f\rightarrow F\right\}$

$\begin{aligned} f= & f\left(\boldsymbol{x}\right)\\ = & f\left(x_{{\scriptscriptstyle 1}},\cdots,x_{{\scriptscriptstyle j}},\cdots,x_{{\scriptscriptstyle n}}\right)\\ = & f\left(x^{{\scriptscriptstyle 1}},\cdots,x^{{\scriptscriptstyle j}},\cdots,x^{{\scriptscriptstyle n}}\right) \end{aligned}$

$\boldsymbol{x}=\left\langle x^{{\scriptscriptstyle 1}},\cdots,x^{{\scriptscriptstyle j}},\cdots,x^{{\scriptscriptstyle n}}\right\rangle$

$\boldsymbol{x}\left(t\right)=\left\langle x^{{\scriptscriptstyle 1}}\left(t\right),\cdots,x^{{\scriptscriptstyle j}}\left(t\right),\cdots,x^{{\scriptscriptstyle n}}\left(t\right)\right\rangle$

$\begin{aligned} \dfrac{\mathrm{d}f}{\mathrm{d}t}= & \dfrac{\mathrm{d}x^{{\scriptscriptstyle 1}}}{\mathrm{d}t}\dfrac{\partial f}{\partial x^{{\scriptscriptstyle 1}}}+\cdots+\dfrac{\mathrm{d}x^{{\scriptscriptstyle j}}}{\mathrm{d}t}\dfrac{\partial f}{\partial x^{{\scriptscriptstyle j}}}+\cdots+\dfrac{\mathrm{d}x^{{\scriptscriptstyle n}}}{\mathrm{d}t}\dfrac{\partial f}{\partial x^{{\scriptscriptstyle n}}}\\ = & \cdots+\dfrac{\mathrm{d}x^{{\scriptscriptstyle j}}}{\mathrm{d}t}\dfrac{\partial f}{\partial x^{{\scriptscriptstyle j}}}+\cdots=\dfrac{\mathrm{d}x^{{\scriptscriptstyle j}}}{\mathrm{d}t}\dfrac{\partial f}{\partial x^{{\scriptscriptstyle j}}}=\dfrac{\mathrm{d}x^{{\scriptscriptstyle j}}}{\mathrm{d}t}\partial_{{\scriptscriptstyle j}}f \end{aligned}$

$\begin{aligned} V= & \mathrm{span}\left\{ \boldsymbol{e}_{{\scriptscriptstyle 1}},\cdots,\boldsymbol{e}_{{\scriptscriptstyle j}},\cdots,\boldsymbol{e}_{{\scriptscriptstyle n}}\right\} \\ = & \mathrm{span}\left\{ \dfrac{\partial}{\partial x^{{\scriptscriptstyle 1}}}|_{{\scriptscriptstyle P}},\cdots,\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}},\cdots,\dfrac{\partial}{\partial x^{{\scriptscriptstyle n}}}|_{{\scriptscriptstyle P}}\right\} \\ = & \mathrm{span}\left\{ \boldsymbol{\partial}_{{\scriptscriptstyle 1}},\cdots,\boldsymbol{\partial}_{{\scriptscriptstyle j}},\cdots,\boldsymbol{\partial}_{{\scriptscriptstyle n}}\right\} \\ = & \left\{ \boldsymbol{\partial}_{{\scriptscriptstyle t}}\middle|\boldsymbol{\partial}_{{\scriptscriptstyle t}}=a_{{\scriptscriptstyle j}}\boldsymbol{e}_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle j}}\boldsymbol{\partial}_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle j}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\right\} \\ = & \left\{ \dfrac{\partial}{\partial t}|_{{\scriptscriptstyle P}}\middle|\dfrac{\partial}{\partial t}|_{{\scriptscriptstyle P}}=a_{{\scriptscriptstyle 1}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle 1}}}|_{{\scriptscriptstyle P}}+\cdots+a_{{\scriptscriptstyle j}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}+\cdots+a_{{\scriptscriptstyle n}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle n}}}|_{{\scriptscriptstyle P}}\right\} \end{aligned}$

44.2 dual space of span of partials

$V^{*}=\left\{ \boldsymbol{\omega}_{{\scriptscriptstyle f}}\middle|\boldsymbol{\omega}_{{\scriptscriptstyle f}}:V\rightarrow F\right\}$

$\boldsymbol{\omega}_{{\scriptscriptstyle f}}\left(\boldsymbol{e}_{{\scriptscriptstyle j}}\right)=\boldsymbol{\omega}_{{\scriptscriptstyle f}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=\boldsymbol{\omega}_{{\scriptscriptstyle f}}\left(\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\right)=\dfrac{\partial f}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\in F$

$\begin{aligned} \boldsymbol{\omega}_{{\scriptscriptstyle fg}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=\dfrac{\partial fg}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}= & f|_{{\scriptscriptstyle P}}\dfrac{\partial g}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}+\dfrac{\partial f}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}g|_{{\scriptscriptstyle P}}\\ = & f|_{{\scriptscriptstyle P}}\boldsymbol{\omega}_{{\scriptscriptstyle g}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)+\boldsymbol{\omega}_{{\scriptscriptstyle f}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)g|_{{\scriptscriptstyle P}} \end{aligned}$

$\boldsymbol{\omega}_{{\scriptscriptstyle x}^{{\scriptscriptstyle i}}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=\boldsymbol{\omega}_{{\scriptscriptstyle x}^{{\scriptscriptstyle i}}}\left(\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\right)=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}=\delta_{{\scriptscriptstyle ij}}=\begin{cases} 1 & i=j\\ 0 & i\ne j \end{cases}$

$\begin{aligned} V^{*}= & \left\{ \boldsymbol{\omega}_{{\scriptscriptstyle f}}\middle|\boldsymbol{\omega}_{{\scriptscriptstyle f}}:V\rightarrow F\right\} =\left\{ \boldsymbol{\omega}_{{\scriptscriptstyle f}}\middle|\begin{cases} \boldsymbol{\omega}_{{\scriptscriptstyle f}}\left(\boldsymbol{e}_{{\scriptscriptstyle j}}\right)=\boldsymbol{\omega}_{{\scriptscriptstyle f}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=\boldsymbol{\omega}_{{\scriptscriptstyle f}}\left(\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\right)=\dfrac{\partial f}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\in F\\ \boldsymbol{\omega}_{{\scriptscriptstyle fg}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=f|_{{\scriptscriptstyle P}}\boldsymbol{\omega}_{{\scriptscriptstyle g}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)+\boldsymbol{\omega}_{{\scriptscriptstyle f}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)g|_{{\scriptscriptstyle P}}\\ \boldsymbol{\omega}_{{\scriptscriptstyle x}^{{\scriptscriptstyle i}}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=\boldsymbol{\omega}_{{\scriptscriptstyle x}^{{\scriptscriptstyle i}}}\left(\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\right)=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}=\delta_{{\scriptscriptstyle ij}}=\begin{cases} 1 & i=j\\ 0 & i\ne j \end{cases} \end{cases}\right\} \\ = & \left\{ \mathrm{d}f\middle|\mathrm{d}f:V\rightarrow F\right\} =\left\{ \mathrm{d}f\middle|\begin{cases} \mathrm{d}f\left(\boldsymbol{e}_{{\scriptscriptstyle j}}\right)=\mathrm{d}f\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=\mathrm{d}f\left(\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\right)=\dfrac{\partial f}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\in F\\ \mathrm{d}fg\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=f|_{{\scriptscriptstyle P}}\left(\mathrm{d}g\right)+\left(\mathrm{d}f\right)g|_{{\scriptscriptstyle P}}\\ \mathrm{d}x^{{\scriptscriptstyle i}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=\mathrm{d}x^{{\scriptscriptstyle i}}\left(\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\right)=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}=\delta_{{\scriptscriptstyle ij}}=\begin{cases} 1 & i=j\\ 0 & i\ne j \end{cases} \end{cases}\right\} \end{aligned}$

$\mathrm{d}x^{{\scriptscriptstyle i}}\left(\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\right)=\delta_{{\scriptscriptstyle ij}}=\boldsymbol{e}^{{\scriptscriptstyle i}}\cdot\boldsymbol{e}_{{\scriptscriptstyle j}}\Rightarrow\begin{cases} \boldsymbol{e}^{{\scriptscriptstyle i}}=\mathrm{d}x^{{\scriptscriptstyle i}}\\ \boldsymbol{e}_{{\scriptscriptstyle j}}=\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}} \end{cases}$

$\begin{aligned} V^{*}= & \left\{ \mathrm{d}f\middle|\mathrm{d}f:V\rightarrow F\right\} =\left\{ \mathrm{d}f\middle|\begin{cases} \mathrm{d}f\left(\boldsymbol{e}_{{\scriptscriptstyle j}}\right)=\mathrm{d}f\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=\mathrm{d}f\left(\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\right)=\dfrac{\partial f}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\in F\\ \mathrm{d}fg\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=f|_{{\scriptscriptstyle P}}\left(\mathrm{d}g\right)+\left(\mathrm{d}f\right)g|_{{\scriptscriptstyle P}}\\ \mathrm{d}x^{{\scriptscriptstyle i}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=\mathrm{d}x^{{\scriptscriptstyle i}}\left(\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\right)=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}=\delta_{{\scriptscriptstyle ij}}=\begin{cases} 1 & i=j\\ 0 & i\ne j \end{cases} \end{cases}\right\} \\ = & \mathrm{span}\left\{ \mathrm{d}x^{{\scriptscriptstyle 1}},\cdots,\mathrm{d}x^{{\scriptscriptstyle i}},\cdots,\mathrm{d}x^{{\scriptscriptstyle n}}\right\} =\mathrm{span}\left\{ \boldsymbol{e}^{{\scriptscriptstyle 1}},\cdots,\boldsymbol{e}^{{\scriptscriptstyle j}},\cdots,\boldsymbol{e}^{{\scriptscriptstyle n}}\right\} \end{aligned}$

44.3 directional derivative

$\begin{aligned} \mathrm{d}f\left(\boldsymbol{v}\right)= & \mathrm{d}f\left(v_{{\scriptscriptstyle j}}\boldsymbol{e}_{{\scriptscriptstyle j}}\right)=v_{{\scriptscriptstyle j}}\mathrm{d}f\left(\boldsymbol{e}_{{\scriptscriptstyle j}}\right)\\ = & v_{{\scriptscriptstyle j}}\mathrm{d}f\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=v_{{\scriptscriptstyle j}}\dfrac{\partial f}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\\ = & v_{{\scriptscriptstyle 1}}\dfrac{\partial f}{\partial x^{{\scriptscriptstyle 1}}}|_{{\scriptscriptstyle P}}+\cdots+v_{{\scriptscriptstyle j}}\dfrac{\partial f}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}+\cdots+v_{{\scriptscriptstyle n}}\dfrac{\partial f}{\partial x^{{\scriptscriptstyle n}}}|_{{\scriptscriptstyle P}}\\ = & \begin{pmatrix}v_{{\scriptscriptstyle 1}} & \cdots & v_{{\scriptscriptstyle j}} & \cdots & v_{{\scriptscriptstyle n}}\end{pmatrix}\boldsymbol{\nabla}f \end{aligned}$

$\overset{\frown}{PQ}=C\left(t\right)-C\left(0\right)=Q-P$

$\boldsymbol{v}=\dfrac{\partial}{\partial t}|_{{\scriptscriptstyle P}}$

$\begin{aligned} \mathrm{d}f\left(s\boldsymbol{v}\right)= & \mathrm{d}f\left(s\dfrac{\partial}{\partial t}|_{{\scriptscriptstyle P}}\right)=s\dfrac{\partial f}{\partial t}|_{{\scriptscriptstyle P}}\\ = & s\boldsymbol{v}\left(f\right)=s\cdot\lim_{t\rightarrow0}\dfrac{f\left(C\left(t\right)\right)-f\left(C\left(0\right)\right)}{t}\\ \approx & s\cdot\dfrac{f\left(Q\right)-f\left(P\right)}{s}=f\left(Q\right)-f\left(P\right)=\Delta f \end{aligned}$

44.4 coefficient of linear combination for vector space and dual space

$\begin{aligned} V= & \left\{ \boldsymbol{v}\middle|\boldsymbol{v}:f\rightarrow F\right\} \\ = & \mathrm{span}\left\{ \boldsymbol{e}_{{\scriptscriptstyle 1}},\cdots,\boldsymbol{e}_{{\scriptscriptstyle j}},\cdots,\boldsymbol{e}_{{\scriptscriptstyle n}}\right\} \\ = & \mathrm{span}\left\{ \dfrac{\partial}{\partial x^{{\scriptscriptstyle 1}}}|_{{\scriptscriptstyle P}},\cdots,\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}},\cdots,\dfrac{\partial}{\partial x^{{\scriptscriptstyle n}}}|_{{\scriptscriptstyle P}}\right\} =\mathrm{span}\left\{ \boldsymbol{\partial}_{{\scriptscriptstyle 1}},\cdots,\boldsymbol{\partial}_{{\scriptscriptstyle j}},\cdots,\boldsymbol{\partial}_{{\scriptscriptstyle n}}\right\} \\ = & \left\{ \boldsymbol{\partial}_{{\scriptscriptstyle t}}\middle|\boldsymbol{\partial}_{{\scriptscriptstyle t}}=a_{{\scriptscriptstyle j}}\boldsymbol{e}_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle j}}\boldsymbol{\partial}_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle j}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\right\} \\ = & \left\{ \dfrac{\partial}{\partial t}|_{{\scriptscriptstyle P}}\middle|\dfrac{\partial}{\partial t}|_{{\scriptscriptstyle P}}=a_{{\scriptscriptstyle 1}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle 1}}}|_{{\scriptscriptstyle P}}+\cdots+a_{{\scriptscriptstyle j}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}+\cdots+a_{{\scriptscriptstyle n}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle n}}}|_{{\scriptscriptstyle P}}\right\} \\ V^{*}= & \left\{ \mathrm{d}f\middle|\mathrm{d}f:V\rightarrow F\right\} \\ = & \mathrm{span}\left\{ \boldsymbol{e}^{{\scriptscriptstyle 1}},\cdots,\boldsymbol{e}^{{\scriptscriptstyle i}},\cdots,\boldsymbol{e}^{{\scriptscriptstyle n}}\right\} \\ = & \mathrm{span}\left\{ \mathrm{d}x^{{\scriptscriptstyle 1}},\cdots,\mathrm{d}x^{{\scriptscriptstyle i}},\cdots,\mathrm{d}x^{{\scriptscriptstyle n}}\right\} \\ = & \left\{ \mathrm{d}f\middle|\mathrm{d}f=b^{{\scriptscriptstyle i}}\boldsymbol{e}^{{\scriptscriptstyle i}}=b^{{\scriptscriptstyle i}}\mathrm{d}x^{{\scriptscriptstyle i}}\right\} \\ = & \left\{ \mathrm{d}f\middle|\mathrm{d}f=b^{{\scriptscriptstyle 1}}\mathrm{d}x^{{\scriptscriptstyle 1}}+\cdots+b^{{\scriptscriptstyle i}}\mathrm{d}x^{{\scriptscriptstyle i}}+\cdots+b^{{\scriptscriptstyle n}}\mathrm{d}x^{{\scriptscriptstyle n}}\right\} \end{aligned}$

or more simply to be comparison

$\begin{array}{ccccccccccc} V & =\mathrm{span}\{ & \boldsymbol{e}_{{\scriptscriptstyle j}}= & \partial_{{\scriptscriptstyle j}} & \}=\{ & \boldsymbol{v}= & \partial_{{\scriptscriptstyle t}}|_{{\scriptscriptstyle P}} & =a_{{\scriptscriptstyle j}}\thinspace\boldsymbol{e}_{{\scriptscriptstyle j}}= & a_{{\scriptscriptstyle j}}\ \partial_{{\scriptscriptstyle j}}|_{{\scriptscriptstyle P}} & :f\rightarrow F & \}\\ V^{*} & =\mathrm{span}\{ & \boldsymbol{e}^{{\scriptscriptstyle i}}= & \mathrm{d}x^{{\scriptscriptstyle i}} & \}=\{ & \boldsymbol{\omega}= & \mathrm{d}f & =b^{{\scriptscriptstyle i}}\thinspace\boldsymbol{e}^{{\scriptscriptstyle i}}= & b^{{\scriptscriptstyle i}}\ \mathrm{d}x^{{\scriptscriptstyle i}} & :V\rightarrow F & \} \end{array}$

$\begin{aligned} & \begin{cases} \mathrm{d}x^{{\scriptscriptstyle i}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=\mathrm{d}x^{{\scriptscriptstyle i}}\left(\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}\right)=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}=\delta_{{\scriptscriptstyle ij}}=\begin{cases} 1 & i=j\\ 0 & i\ne j \end{cases}\\ \boldsymbol{\partial}_{{\scriptscriptstyle t}}=a_{{\scriptscriptstyle j}}\boldsymbol{e}_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle j}}\boldsymbol{\partial}_{{\scriptscriptstyle j}}\Leftrightarrow\dfrac{\partial}{\partial t}|_{{\scriptscriptstyle P}}=a_{{\scriptscriptstyle 1}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle 1}}}|_{{\scriptscriptstyle P}}+\cdots+a_{{\scriptscriptstyle j}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle j}}}|_{{\scriptscriptstyle P}}+\cdots+a_{{\scriptscriptstyle n}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle n}}}|_{{\scriptscriptstyle P}} \end{cases}\\ \Rightarrow & \begin{cases} \mathrm{d}x^{{\scriptscriptstyle i}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle t}}\right)=\mathrm{d}x^{{\scriptscriptstyle i}}\left(\dfrac{\partial}{\partial t}|_{{\scriptscriptstyle P}}\right)=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial t}|_{{\scriptscriptstyle P}}\\ \mathrm{d}x^{{\scriptscriptstyle i}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle t}}\right)=\mathrm{d}x^{{\scriptscriptstyle i}}\left(a_{{\scriptscriptstyle j}}\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=a_{{\scriptscriptstyle j}}\mathrm{d}x^{{\scriptscriptstyle i}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=a_{{\scriptscriptstyle j}}\delta_{{\scriptscriptstyle ij}}=a_{{\scriptscriptstyle i}} \end{cases}\Rightarrow a_{{\scriptscriptstyle i}}=\mathrm{d}x^{{\scriptscriptstyle i}}\left(\boldsymbol{\partial}_{{\scriptscriptstyle t}}\right)=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial t}|_{{\scriptscriptstyle P}}\\ \Rightarrow & a_{{\scriptscriptstyle i}}=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial t}|_{{\scriptscriptstyle P}}\Rightarrow a_{{\scriptscriptstyle j}}=\dfrac{\partial x^{{\scriptscriptstyle j}}}{\partial t}|_{{\scriptscriptstyle P}}=\partial_{{\scriptscriptstyle t}}x^{{\scriptscriptstyle j}}|_{{\scriptscriptstyle P}}\\ \Rightarrow & \dfrac{\partial}{\partial t}|_{{\scriptscriptstyle P}}=a_{{\scriptscriptstyle i}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle i}}}|_{{\scriptscriptstyle P}}=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial t}|_{{\scriptscriptstyle P}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle i}}}|_{{\scriptscriptstyle P}}=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial t}\dfrac{\partial}{\partial x^{{\scriptscriptstyle i}}}|_{{\scriptscriptstyle P}}\Rightarrow\dfrac{\partial}{\partial t}=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial t}\dfrac{\partial}{\partial x^{{\scriptscriptstyle i}}}\\ \Rightarrow & \partial_{{\scriptscriptstyle t}}|_{{\scriptscriptstyle P}}=\dfrac{\partial x^{{\scriptscriptstyle j}}}{\partial t}\partial_{{\scriptscriptstyle j}}|_{{\scriptscriptstyle P}}\Leftrightarrow\partial_{{\scriptscriptstyle t}}|_{{\scriptscriptstyle P}}=\partial_{{\scriptscriptstyle t}}x^{{\scriptscriptstyle j}}\partial_{{\scriptscriptstyle j}}|_{{\scriptscriptstyle P}} \end{aligned}$

$\begin{aligned} \mathrm{d}f= & b^{{\scriptscriptstyle i}}\boldsymbol{e}^{{\scriptscriptstyle i}}=b^{{\scriptscriptstyle i}}\mathrm{d}x^{{\scriptscriptstyle i}}\\ \dfrac{\partial f}{\partial x^{{\scriptscriptstyle j}}}=\mathrm{d}f\left(\boldsymbol{\partial}_{{\scriptscriptstyle j}}\right)=\mathrm{d}f\left(\boldsymbol{e}_{{\scriptscriptstyle j}}\right)= & b^{{\scriptscriptstyle i}}\boldsymbol{e}^{{\scriptscriptstyle i}}\cdot\boldsymbol{e}_{{\scriptscriptstyle j}}=b^{{\scriptscriptstyle i}}\delta_{{\scriptscriptstyle ij}}=b^{{\scriptscriptstyle j}}\\ b^{{\scriptscriptstyle j}}= & \dfrac{\partial f}{\partial x^{{\scriptscriptstyle j}}}\\ b^{{\scriptscriptstyle i}}= & \dfrac{\partial f}{\partial x^{{\scriptscriptstyle i}}}=\partial_{{\scriptscriptstyle i}}f\\ \mathrm{d}f=b^{{\scriptscriptstyle i}}\boldsymbol{e}^{{\scriptscriptstyle i}}=b^{{\scriptscriptstyle i}}\mathrm{d}x^{{\scriptscriptstyle i}}= & \dfrac{\partial f}{\partial x^{{\scriptscriptstyle i}}}\mathrm{d}x^{{\scriptscriptstyle i}}\\ \mathrm{d}f= & \dfrac{\partial f}{\partial x^{{\scriptscriptstyle i}}}\mathrm{d}x^{{\scriptscriptstyle i}}\\ \mathrm{d}f= & \partial_{{\scriptscriptstyle i}}f\mathrm{d}x^{{\scriptscriptstyle i}} \end{aligned}$

$\begin{array}{ccccccccccc} V & =\mathrm{span}\{ & \boldsymbol{e}_{{\scriptscriptstyle j}}= & \partial_{{\scriptscriptstyle j}} & \}=\{ & \boldsymbol{v}= & \partial_{{\scriptscriptstyle t}}|_{{\scriptscriptstyle P}} & =a_{{\scriptscriptstyle j}}\thinspace\boldsymbol{e}_{{\scriptscriptstyle j}}= & \partial_{{\scriptscriptstyle t}}x^{{\scriptscriptstyle j}}\ \partial_{{\scriptscriptstyle j}}|_{{\scriptscriptstyle P}} & :f\rightarrow F & \}\\ V^{*} & =\mathrm{span}\{ & \boldsymbol{e}^{{\scriptscriptstyle i}}= & \mathrm{d}x^{{\scriptscriptstyle i}} & \}=\{ & \boldsymbol{\omega}= & \mathrm{d}f & =b^{{\scriptscriptstyle i}}\thinspace\boldsymbol{e}^{{\scriptscriptstyle i}}= & \partial_{{\scriptscriptstyle i}}f\ \mathrm{d}x^{{\scriptscriptstyle i}} & :V\rightarrow F & \} \end{array}$

44.5 change of basis / change of coordinate

$\dfrac{\partial}{\partial t}=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial t}\dfrac{\partial}{\partial x^{{\scriptscriptstyle i}}}\overset{t=x^{\prime{\scriptscriptstyle j}}}{\Rightarrow}\dfrac{\partial}{\partial x^{\prime{\scriptscriptstyle j}}}=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial x^{\prime{\scriptscriptstyle j}}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle i}}}=\dfrac{\partial x^{{\scriptscriptstyle 1}}}{\partial x^{\prime{\scriptscriptstyle j}}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle 1}}}+\dfrac{\partial x^{{\scriptscriptstyle 2}}}{\partial x^{\prime{\scriptscriptstyle j}}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle 2}}}+\dfrac{\partial x^{{\scriptscriptstyle 3}}}{\partial x^{\prime{\scriptscriptstyle j}}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle 3}}}$

$\begin{aligned} \mathrm{d}f= & \dfrac{\partial f}{\partial x^{{\scriptscriptstyle i}}}\mathrm{d}x^{{\scriptscriptstyle i}}\\ f=x^{\prime{\scriptscriptstyle j}}\Downarrow\\ \mathrm{d}x^{\prime{\scriptscriptstyle j}}= & \dfrac{\partial x^{\prime{\scriptscriptstyle j}}}{\partial x^{{\scriptscriptstyle i}}}\mathrm{d}x^{{\scriptscriptstyle i}} \end{aligned}$

$\begin{cases} \dfrac{\partial}{\partial x^{\prime{\scriptscriptstyle j}}}=\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial x^{\prime{\scriptscriptstyle j}}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle i}}}=\sum\limits _{i}\dfrac{\partial x^{{\scriptscriptstyle i}}}{\partial x^{\prime{\scriptscriptstyle j}}}\dfrac{\partial}{\partial x^{{\scriptscriptstyle i}}}\\ \mathrm{d}x^{\prime{\scriptscriptstyle j}}=\dfrac{\partial x^{\prime{\scriptscriptstyle j}}}{\partial x^{{\scriptscriptstyle i}}}\mathrm{d}x^{{\scriptscriptstyle i}}=\sum\limits_{i}\dfrac{\partial x^{\prime{\scriptscriptstyle j}}}{\partial x^{{\scriptscriptstyle i}}}\mathrm{d}x^{{\scriptscriptstyle i}} \end{cases}$