Terminologies in Differential Calculus

• Differential

  Linear map $df_p: \mathbb{R}^n \rightarrow \mathbb{R}$ on tangent vector

  \[\underbrace{v\rightarrow df_p(v)}_\texttt{linear} ,\; \underbrace{p\rightarrow df_p(v)}_\texttt{(possibly) non-liear}\]
  • Tangent vector: 어떤 점이 주어졌을 때, 어떤 방향으로 이동(displacement)할 것인가?
  • Tangent vector에 대한 linear map이기 때문에 cotangent vector로 볼 수 있다.
  • $p$ 마다 linear map이 존재하니까, p에 대해서는 vector field로 볼 수 있다.

  • 그렇다면 linear map의 basis는 무엇인가? Coordinate vector field

    \[\left\{(dx^i)_p := \left(\frac{\partial}{\partial x^i}\right)_p\right\}_{i=1}^{n}\]

  • 임의의 differential에 대해서 Coordinate vector field로 표현된 형태는 다음과 같다.

    \[df_p= \sum_{i=1}^{n}\left(\frac{\partial f}{\partial x^i}\right)_p (dx^i)_p\]

    즉, partial derivative $\frac{\partial f}{\partial x^i}$는 differential의 좌표의 역할을 한다. 또한 위 식은 각각의 partial derivative가 $p$에 대한 scalar field, coordinate vector field가 $p$에 대한 vector field로 정의된다.

• Pushforward

  Linear approximation of a smooth map $\varphi: M \rightarrow N$ between tangent space $d\varphi: TM \rightarrow TN$. Using commutative diagram, we have

  \[\require{AMScd} \begin{CD} TM @>d\varphi>> TN\\ @V\pi_MVV @V\pi_NVV\\ M @>\varphi>> N \end{CD}\]
  • $\pi_M, \pi_N$: Bundle projection
  • Jacobian of $\varphi$ at x: Matrix representation of $d\varphi_x$

• Differential form

  provide a framework which accomodates multiplication and differentiation of differentials

  • 각 tangent space $T_pM$에 대해서 linear 하다. (i.e., multilinear or Tensor)

    \[w_p: \wedge^k T_pM \rightarrow \mathbb{R}\]
    • Tangent space의 위치를 서로 바꿨을 때는 sign이 바뀐다.(i.e., alternative)
    • In summary, Alternative tensor

  • Multiplication: Exterior product = Differential form끼리 곱해서 새로운 Differential form을 만드는 연산

    \[\wedge: \Omega^k(M) \times \Omega^l(M) \rightarrow \Omega^{k+l}(M)\]

  • Differentiation: Exterior derivative = Differential form을 미분해서 새로운 Differential form을 만드는 연산

    \[d:\Omega^k(M) \rightarrow \Omega^{k+1}(M)\]

    For k-form $\varphi \in \Omega^k(M)$, we have following representation

    \[\varphi = \sum_{I \in \mathcal{J}_{k,n}} f_I dx^I\]

    Then, the exterior derivative $d\varphi \in \Omega^{k+1}(M)$ of $\varphi$ is

    \[d\varphi = \sum_{I \in \mathcal{J}_{k,n}}\overbrace{\underbrace{\sum_{i=1}^{n}\frac{\partial f_I}{\partial x^i}dx^i}_{df_I} \wedge dx^I}^{df_I \wedge dx^I}\]

    물론, 위 식은 pointwise하게 정의된다.

    \[d\varphi_p = \sum_{I \in \mathcal{J}_{k,n}}\sum_{i=1}^{n}\left(\frac{\partial f_I}{\partial x^i}\right)_p \left(dx^i \wedge dx^I\right)_p\]

  • Application: Del series are differtial forms!

    • Gradient: Let us assume 0-form (scalar field) $f: \mathbb{R}^n \rightarrow \mathbb{R}$. Then, the exterior derivative of $f$ is 1-form

      \[df = \sum_{i=1}^{n}\frac{\partial f}{\partial x^i}dx^i\]

    • Curl: Let us assume 1-form $\omega = A_x dx + A_ydy + A_z dz$. Then, the exterior derivative of $\omega$ is 2-form

      \[d\omega = \frac{\partial A_x}{\partial y} dy \wedge dx + \frac{\partial A_x}{\partial z} dz \wedge dx + \frac{\partial A_y}{\partial x} dx \wedge dy + \frac{\partial A_y}{\partial z} dz \wedge dy + \frac{\partial A_z}{\partial x} dx \wedge dz + \frac{\partial A_z}{\partial y} dy \wedge dz \\ =\left(\frac{\partial A_y}{\partial x}- \frac{\partial A_x}{\partial y}\right)dx \wedge dy + \left(\frac{\partial A_z}{\partial y}- \frac{\partial A_y}{\partial z}\right)dy\wedge dz + \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right)dz \wedge dx\]

    • Divergence: Let us assume 2-form $\varphi: B_z dx\wedge dy + B_x dy\wedge dz + B_y dz \wedge dx$. Then, the exterior derivative of $\varphi$ is 3-form

      \[d\varphi = \frac{\partial B_z}{\partial z}dz\wedge dx \wedge dy + \frac{\partial B_x}{\partial x}dx \wedge dy \wedge dz + \frac{\partial B_y}{\partial y}dy \wedge dz \wedge dx = \left( \frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} + \frac{\partial B_z}{\partial z} \right)dx \wedge dy \wedge dz\]

• Pullback of differential

  Let us assume $f: M \rightarrow N$ is smooth and $\omega \in \Omega^k(N)$ is a k-form on manifold $N$. Then we can naturally define a k-form on manifold $M$, named pullback of $\omega$ on $M$.

  \[f^*\omega_p(v_1, \dots, v_k) = \omega_{f(p)}(f_*(v_1), \dots, f_*(v_k)) = \omega_{f(p)}(df_p(v_1), \dots, df_p(v_k))\]

  where

  \[v_i \in T_pM,\; \forall i=1,\dots,k\]

  Then,

  \[f_*(v_i) = df_p(v_i) \in T_pN\]

  as $df_p$ is a pushforward from $T_pM$ to $T_PN$.

• Pushforward and pullback of measure

  Similar to the pullback of differential, we can pushforward/pullback measure to image space of function. Let us assume $T:M \rightarrow N$ is smooth and $\mu$ is a Lebesgue measure on $M$ and $\nu$ is a Lebesgue measure on $N$. Then, we can define pushforward $\mu$ and pullback $\nu$ as follows

  \[T_*\mu = \mu(T^{-1}(A)),\;\forall A \in \Sigma_M\\ T^*\nu = \nu(T(A)),\;\forall A \in \Sigma_N\]
  • Change-of-variables formula (Reparameterization trick): We can integrate over manifold $N$ by integrating over manifold $M$. Let us assume $g:M \rightarrow \mathbb{R}$ and $h:N \rightarrow \mathbb{R}$ is a measurable functions. Then we have

    • Reparameterization trick for pushforward measure

      \[\int_\Omega g d\mu = \int_{T(\Omega)}g\circ T^{-1} dT_*\mu = \int_{T(\Omega)}g\circ T^{-1}\frac{dT_*\mu}{d\nu}d\nu = \int_{T(\Omega)}g\circ T^{-1}|D_xT^{-1}|d\nu\]

    • Reparameterization trick for pullback measure

      \[\int_{T(\Omega)}hd\nu = \int_{\Omega}h\circ TdT^*\nu = \int_{\Omega}h\circ T \frac{dT^*\nu}{d\mu}d\mu = \int_{\Omega}h\circ T |D_xT| d\mu\]