diff --git a/content/Math401/Extending_thesis/Math401_S5.md b/content/Math401/Extending_thesis/Math401_S5.md index a69949d..a5c7e97 100644 --- a/content/Math401/Extending_thesis/Math401_S5.md +++ b/content/Math401/Extending_thesis/Math401_S5.md @@ -22,8 +22,14 @@ What is form means here? ### Differential Forms in our sweet home, $\mathbb{R}^n$ +> [!NOTE] +> +> I'm a bit deviated form the notation we used in the book, in the actual text, they use $\mathbb{R}^n_p$ to represent the tangent space of $\mathbb{R}^n$ at $p$. But to help you link those concepts as we see in smooth manifolds, $T_pM$, we will use $T_p\mathbb{R}^n$ to represent the tangent space of $\mathbb{R}^n$ at $p$. + Let $p$ be a point in $\mathbb{R}^n$. The tangent space of $\mathbb{R}^n$ at $p$ is denoted by $T_p\mathbb{R}^n$, is the set of all vectors in $\mathbb{R}^n$ that use $p$ as origin. +#### Definition of a vector field + A vector field is a map that associates to each point $p$ in $\mathbb{R}^n$ a vector $v(p)$ in $T_p\mathbb{R}^n$. That is @@ -40,3 +46,192 @@ $v$ is differentiable at $p$ if the function $a_i$ is differentiable at $p$. This gives a vector field $v$ on $\mathbb{R}^n$. +#### Definition of dual space of tangent space + +To each tangent space $T_p\mathbb{R}^n$ we can associate the dual space $(T_p\mathbb{R}^n)^*$, the set of all linear maps from $T_p\mathbb{R}^n$ to $\mathbb{R}$. ($\varphi:T_p\mathbb{R}^n\to \mathbb{R}$) + +The basis for $(T_p\mathbb{R}^n)^*$ is obtained by taking $(dx_i)_p$ for $i=1,...,n$. + +This is the dual basis for $\{(e_i)_p\}$ since. + +$$ +(dx_i)_p(e_j)=\frac{\partial x_i}{\partial x_j}=\begin{cases}0 \text{ if } i\neq j\\ +1 \text{ if } i=j +\end{cases} +$$ + +#### Definition of a 1-form + +A 1-form is a linear map from $(T_p\mathbb{R}^n)^*$ to $\mathbb{R}$. + +$$ +\omega(p)=a_1(p)(dx_1)_p+...+a_n(p)(dx_n)_p +$$ + +where $a_i(p)$ is a function that maps $\mathbb{R}^n$ to $\mathbb{R}$. + +Generalization of 1-form is $k$-form defined as follows: + +#### Definition of a $k$-form + +We can define the set of linear map $\Lambda^2(\mathbb{R}^n_p)^*$ where $\varphi$ maps from $(T_p\mathbb{R}^n)^*\times ... \times (T_p\mathbb{R}^n)^*$ to $\mathbb{R}$, that are bilinear and alternate ($\varphi(v_1,v_2)=-\varphi(v_2,v_1$). + +when $\varphi_1$ and $\varphi_2$ are linear maps from $(T_p\mathbb{R}^n)^*$ to $\mathbb{R}$, then $\varphi_1\wedge \varphi_2$ is a bilinear map from $(T_p\mathbb{R}^n)^*\times (T_p\mathbb{R}^n)^*$ to $\mathbb{R}$ by setting + +$$ +(\varphi_1\wedge \varphi_2)(v_1,v_2)=\varphi_1(v_1)\varphi_2(v_2)-\varphi_1(v_2)\varphi_2(v_1)=\det(\varphi_i(v_j)) +$$ + +where $i,j=1,\ldots,k$, $k$ is the degree of the exterior form + +More generally, $(\varphi_1\wedge \varphi_2\wedge\dots \wedge \varphi_k)(v_1,v_2,\dots,v_k)=\det(\varphi_i(v_j))$. + +And $\{(dx_i\wedge dx_j)_p,i +Example for real space 4 product + +0-forms: functino in $\mathbb{R}^4$ + +1-forms: $a_1(p)(dx_1)_p+a_2(p)(dx_2)_p+a_3(p)(dx_3)_p+a_4(p)(dx_4)_p$ + +2-forms: $a_{12}(p)(dx_1\wedge dx_2)_p+a_{13}(p)(dx_1\wedge dx_3)_p+a_{14}(p)(dx_1\wedge dx_4)_p+a_{23}(p)(dx_2\wedge dx_3)_p+a_{24}(p)(dx_2\wedge dx_4)_p+a_{34}(p)(dx_3\wedge dx_4)_p$ + +3-forms: $a_{123}(p)(dx_1\wedge dx_2\wedge dx_3)_p+a_{124}(p)(dx_1\wedge dx_2\wedge dx_4)_p+a_{134}(p)(dx_1\wedge dx_3\wedge dx_4)_p+a_{234}(p)(dx_2\wedge dx_3\wedge dx_4)_p$ + +4-forms: $a_{1234}(p)(dx_1\wedge dx_2\wedge dx_3\wedge dx_4)_p$ + + +#### Exterior product of forms + +Let $\omega=\sum a_{I}dx_I$ be a k form where $I=(i_1,i_2,\ldots,i_k)$ and $i_1 +Example for exterior product of forms + +Let $\omega=x_1dx_1+x_2dx_2+x_3dx_3$ be a 1-form in $\mathbb{R}^3$ and $\varphi=x_1dx_1\wedge dx_1\wedge dx_3$ be a 2-form in $\mathbb{R}^3$. + +Then + +$$ +\begin{aligned} +\omega\wedge \varphi&=x_2 dx_2\wedge dx_1\wedge dx_3+x_3x_1 dx_3\wedge dx_1\wedge dx_2\\ +&=(x_1x_3-x_2)dx_1\wedge dx_2\wedge dx_3 +\end{aligned} +$$ + +Note $dx_1\wedge dx_1=0$ therefore $dx_1\wedge dx_1\wedge dx_3=0$ + + +#### Additional properties of exterior product + +Let $\omega$ be a $k$ form, $\varphi$ be a $s$ form, and $\theta$ be an $r$ form. + +- $(\omega\wedge\varphi)\wedge\theta=\omega\wedge(\varphi\wedge\theta)$ +- $(\omega\wedge\varphi)=(-1)^{k+s}(\varphi\wedge\omega)$ +- $\omega\wedge(\varphi+\theta)=\omega\wedge\varphi+\omega\wedge\theta$ + +#### Important implications with differential maps + +Let $f:\mathbb{R}^n\to \mathbb{R}^m$ be a differentiable map. Then $f$ induces a map $f^*$ from k-forms in $\mathbb{R}^n$ to k-forms in $\mathbb{R}^m$. + +That is + +$$ +(f^*\omega)(p)(v_1,\ldots,v_k)=\omega(f(p))(df(p)_1v_1,\ldots,df(p)_kv_k) +$$ + +Here $p\in \mathbb{R}^n$, $v_1,\ldots,v_k\in T_p\mathbb{R}^n$, and $df(p):T_p\mathbb{R}^n\to T_{f(p)}\mathbb{R}^m$. + +If $g$ is a 0-form, we have + +$f^*(g)=g\circ f$ + +#### Additional properties for differential maps + +Let $f:\mathbb{R}^n\to \mathbb{R}^m$ be a differentiable map, $\omega,\varphi$ be k-forms on $\mathbb{R}^m$ and $g:\mathbb{R}^m\to \mathbb{R}$ be a 0-form on $\mathbb{R}^m$. Then: + +- $f^*(\omega+\varphi)=f^*\omega+f^*\varphi$ +- $f^*(g\omega)=f^*(g)f^*\omega$ +- If $\varphi_1,\dots,\varphi_k$ are 1-forms in $\mathbb{R}^m$, $f^*(\varphi_1\wedge\cdots\wedge\varphi_k)=f^*\varphi_1\wedge\cdots\wedge f^*\varphi_k$ + +If $g:\mathbb{R}^p\to \mathbb{R}^n$ is a differential map and $\varphi,\omega$ are any two-forms in $\mathbb{R}^m$. + +- $f^*(\omega\wedge\varphi)=f^*\omega\wedge f^*\varphi$ +- $(f\circ g)^*omega=g^*(f^*\omega)$ + +#### Exterior Differential + +Let $\omega=\sum a_{I}dx_I$ be a k form in $mathbb{R}^n$. The exterior differential $d\omega$ of $\omega$ is defined by + +$$ +d\omega=\sum da_{I}\wedge dx_I +$$ + +#### Additional properties of exterior differential + +- $d(\omega_1+\omega_2)=d\omega_1+d\omega_2$ where $\omega_1,\omega_2$ are k-forms +- $d(\omega\wedge\varphi)=d\omega\wedge\varphi+(-1)^kw\wedge d\varphi$ where $\omega$ is a k-form and $\varphi$ is a s-form +- $d(d\omega)=d^2\omega=0$ +- $d(f^*\omega)=f^*d\omega$ where $f$ is a differentiable map and $\omega$ is a k-form + +## Differentiable manifolds + +### A different flavor of differential manifolds + +#### Definition of differentiable manifold + +An $n$-dimensional differentiable manifold is a set $M$ together with a family of of injective maps $f_\alpha:U_\alpha\subseteq \mathbb{R}^n\to M$ of open sets $U_\alpha$ in $\mathbb{R}^n$ in to $M$ such that: + +- $\bigcup_\alpha f_\alpha(U_\alpha)=M$ +- For each pair $\alpha,\beta$, with $f_\alpha(U_\alpha)\cap f_\beta(U_\beta)=W\neq \emptyset$, the sets $f_\alpha^{-1}(W)$ and $f_\beta^{-1}(W)$ are open sets in $\mathbb{R}^n$ and the maps $f_\beta^{-1}\circ f_\alpha$ and $f_\alpha^{-1}\circ f_\beta$ are differentiable. +- The family $\{(U_\alpha,f_\alpha)\}$ is the maximal relative to the two properties above. + +> This condition is weaker than smooth manifold, in smooth manifold, we require the function to be class of $C^\infty$ (continuous differentiable of all order), now we only needs it to be differentiable. + +#### Definition of differentiable map between differentiable manifolds + +Let $M_1^n$ and $M_2^M$ be differentiable manifolds. A map $\varphi:M_1\to M_2$ is a differentiable at a point $p\in M_1$ if given a parameterization $g:V\subset \mathbb{R}^m\to M_2$ around $\varphi(p)$, there exists a parameterization $f:U\subseteq \mathbb{R}^n\to M_1$ around $p$ such that: + +$\varphi(f(U))\subset g(V)$ and the map + +$$ +g^{-1}\circ \varphi\circ f: U\subset \mathbb{R}^n\to \mathbb{R}^m +$$ + +is differentiable at $f^{-1}(p)$. + +It is differentiable in an open set of $M_1$ if it is differentiable at all points in such set. + +The map $g^{-1}\circ \varphi\circ f$ is the expression of parameterization of $f$ and $g$. (Since the change of parameterization is differentiable, the property that $f$ is differentiable does not depends on the choice of parameterization.) + +#### Tangent vector over differentiable curve + +Let $\alpha: I\to M$ be a differentiable curve on a differentiable manifold $M$, with $\alpha(0)=p\in M$, and let $D$ be the set of functions of $M$ which are differentiable at $p$. then tangent vector to the curve $\alpha$ at $p$ is the map $\alpha'(0):D\to \mathbb{R}$ given by + +$$ +\alpha'(0)\varphi=\frac{d}{dt}(\varphi\cdot \alpha(t))|_{t=0} +$$ + +A tangent vector at $p\in M$ is the