March 28

  • Consider the \(k\)-form \(\omega \) defined at \(x\in \Omega\subset \mathbb{R}^n\), given by the sum
    $$ \omega(x)= \sum c_I(x) dx_I $$
    where the multi-index
    \( I=\{1\leq i_1\leq i_2 \leq, \ldots, \leq i_k \leq n\} \).
    Each basis element
    \(dx_I:= dx_{i_1} \wedge dx_{i_2} \ldots \wedge dx_{i_k} \)
    is a \(k\)-form taking the (tangent) vectors \((v_1,\ldots,v_k) \), and its values
    is determined by the\(k\times k\) determinant of the \(I\)-rows sliced from
    the \(n\times k\) matrix of stacked column \(v_i\) , i.e
    $$ dx_I(v_1,\ldots,v_k) = \text{det}\left[\begin{array}{c|c|c|c|c} ~&v_1 & v_2 & \ldots & v_k \\
    \hline
    i_1 & & & & \\
    \vdots & & & & \\
    i_n & & & &
    \end{array} \right]$$
  • Integration of forms : we can integrate \(k\)-form over \(k\)-dimensional patches,
    generalizing the idea behind Riemann integration in elementary calculus.
    Recall that the (definite, multivariate) Riemann integral, denoted
    \( \int_{\mathcal{S}} f(x_1,\ldots, x_n) dx_1\cdots dx_n \)
    is a real number involving two main components:

    1. The integration domain \(\mathcal{S} \subset \mathbb{R}^n\)
    2. The integrand \(f(x_1,\ldots, x_n)\).
      The term \(dx_1\cdots dx_n\) symbolizes the generalized infinitesimal area/volume term.
      We can combine those two terms to one \(f(x) dx\).

    By parameterization, this extends to integration over surfaces, curves and volumes, giving rise
    to the beautiful theory of vector calculus.
    This theory can be elegantly generalized to the theory of integrals of differential
    forms

    Our new type of integral, denoted
    $$ \int_{C} \omega = \text{ The integral of the form } \omega \text{ over the manifold } C, $$
    has two main components:

    1. An integration manifold \(C \subset \mathbb{R}^n\): analogous to
      the domain of integration domain \(\mathcal{S}\) in the Riemann integral above.
      Note that we require this manifold to be
      orientable .
    2. A differential form \(\omega\): assigns a “volume” (a real number)
      to the tangent of each point of \(C\), and takes the role of the \(f(x) dx\),
      (or \(dF(x)\) in the Stieltjes integral).
      It must be a \(k\)-form
      (and not some arbitrary \(k\)-linear functional) for it
      vanish on degenerate tangents.

    Note: we require that the dimension of \(C\) equals to the degree of \(\omega\).
    Surface integrals (on two dimensional manifolds) take \(2\)-forms, volume integrals (on three dimensional
    manifolds) take \(3\)-forms, and in general, if the tangent of \(C\subset \mathbb{R}^n\) at point \(x\)
    is \(k\) dimensional (based on its dimension), then \(\omega\) should be a \(k\)-form.

    We first define the integral on a \(k\)-dimensional cube \(C\), mapped by \(\psi\) into \(\Omega \subset
    \mathbb{R}^n\). This cube can be partitioned to many small cubes of size \(\epsilon\), and
    we defined our integral over the image of the cube as the limit of the sum
    $$ \int_{\psi C}\omega = \lim_{\epsilon\to 0} \sum_{\text{cubes}} \epsilon^k \omega(v_1,\ldots,v_k) $$
    where \( v_k:=\frac{\partial \psi}{\partial e_k} \) is the tangent at the direction
    the \(k\)-th entry of \(\psi\).
    To generalize to smooth manifolds , assume that a patch \(C’\) smoothly maps to the
    cube \(C\) through \(\phi\):
    $$ C’ \rightarrow{\phi} C \rightarrow{\psi} \Omega $$
    Then the integral of \(C\) can be pulled back to \(C’\)
    by defining a differential form:
    \( \psi^* \omega = \sum_I \overbrace{c_I(\psi(y))}^{\psi^*c_I} dx_{i_1} \wedge\ldots\wedge dx_{i_k} \),
    (known as a pullback form), such that
    $$ \int_{C’} \psi^* \omega = \int_{\psi C} \omega $$
    If every coordinate \(x\in \Omega \) is an image of \(y\in C\)
    \( x_i = \psi_i(y)\), then the corresponding \(1\)-form is the sum
    \( dx_i = \sum \frac{\partial \psi_i}{\partial y_j} dy_j \).

  • Properties of the integral.
    1. Linearity (in the form):
      \( \int_{C} \alpha \omega_1+\beta\omega_2 = \alpha \int_C \omega_1 + \beta \int_C \omega_2 \)
    2. Change of variables (manifold): \( \int_C \psi^* \omega = \int_{\psi C} \omega \).
    3. Orientation change: shuffling of two coordinates flips the sign
      of the manifold, denoted \(-C\), whose integral naturally, is negative to the one over \(C\):
      $$ \int_{-C} \omega = -\int_C \omega $$
  • Example (circulation integral): let \(C\) be an oriented loop (clock-wise or counter-clockwise),
    and let \(\omega\) be a \(1\)-form.
    The circulation integral of \(\omega\) over \(C\) is defined exactly as
    \( \int_C \omega \).
    Given some parametrization \(\gamma:[0,1]\to C\) with the closed loop condition
    \(\gamma(0) = \gamma(1)\),we have
    $$ \int_C \omega = \int \sum_i c_{i}(\gamma(t)) dx_i \cdot \dot \gamma
    = \sum_i \int c_i(\gamma(t))\dot \gamma_i(t) dt $$
    Let us compute, for example, the circulation of the form
    \( \omega = \frac{x_1dx_2 – x_2dx_1}{x_1^2+x_2^2} \)
    along the CCW circle parametrized by \(\gamma(t):=(r\cos(2\pi t), r\sin(2\pi t))\)
    on \(I=[0,1]\):
    $$ \int_{\gamma I}\omega = \int \frac{ r \cos(2\pi t) r 2\pi \cos(2\pi t) dt
    + r \sin(2\pi t) r 2\pi \sin(2\pi t) dt}{r^2(\cos^2 +\sin^2)}
    = 2\pi \int_I 1 dt = 2\pi $$
    The same will remain true for any loop encircling the origin (in general,
    the value will correspond to the
    winding number
    of the curve).
  • Example (flux integral) : we wish to compute the flux
    of the form
    $$ \omega = \frac{1}{r^3}(x_1 dx_2\wedge dx_3 + x_2 dx_3\wedge dx_1 + x_3 dx_1\wedge dx_2) $$
    through the sphere
    \(C:=\{x_1^2+x_2^2+x_3^2 =r^2 \}\).

    We can re-arrange the form as
    $$ \omega = \frac{(x_1 dx_2 – x_2 dx_1)}{|r|^2}\wedge \frac{dx_3}{|r|} + \frac{ x_3 dx_1\wedge dx_2}{|r|^3} $$
    The left part of this form vanish due to the spherical symmetry, leaving only the
    right side form \(\frac{ x_3 dx_1\wedge dx_2}{|r|^3}\) whose integral over the sphere
    is merely its surface area \( \int_C \omega = 4\pi \).

  • The differentials mapping of forms (exterior derivative). assume \(\omega = a(x,y) dx + b(x,y) dy\)
    and we want to integrate around a small square \(C\) of size \(\epsilon\times \epsilon\).
    We can integrate using the mid-point values of \(\omega\):
    $$ \begin{align}
    \int_C \omega & \approx b(x+\frac{\epsilon}{2},y)\epsilon + a(x,y+\frac{\epsilon}{2})(-\epsilon)
    + b(x-\frac{\epsilon}{2},y)(-\epsilon) +
    a(x,y-\frac{\epsilon}{2})\epsilon \\
    & = \epsilon [
    b(x+\frac{\epsilon}{2},y) -b(x-\frac{\epsilon}{2},y)
    +a(x,y-\frac{\epsilon}{2})-a(x,y+\frac{\epsilon}{2})] \\
    & \approx \epsilon
    \left[ \frac{\partial b}{\partial x} \epsilon – \frac{\partial a}{\partial y} \epsilon
    +o(\epsilon)\right]
    = \epsilon^2\left [\frac{\partial b}{\partial x} -\frac{\partial a}{\partial y}
    \right] + o(\epsilon^2)
    \end{align} $$

    In general, for any \(k\)-form \(\omega\), there exists unique
    \(k+1\)-form (denoted \(d\omega\)) s.t the flux through the boundary
    of a small cube \(C_{\epsilon}^{k+1}\) is
    the integral of \(d\omega\) over the interior of the cube.
    This \(d\omega\), called the
    differential mapping (or just the differential)
    of \(\omega\), admits several properties:

    1. It takes \(k\)-forms into \(k+1\)-forms.
    2. Differential of functions if \(f\) is a \(0\)-form (namely, a scalar field),
      then \(df = \sum_{i} \frac{\partial f}{\partial x_i} dx_i \)
    3. (wedge) Product rule : \(d(\omega_1\wedge \omega_2) = d\omega_2\wedge \omega_2 + (-1)^k\omega_1\wedge d\omega_2 \)
    4. \(d^2 =0 \), namely: \(d(d\omega) =0\)
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