# Exterior Differential forms

$$\def\pd{\partial} \def\Real{\mathbb{R}}$$

• We define a differential $$k$$-form as
$$\omega(x) = \sum_{I=\{i_1<i_2<\ldots<i_k\}} c_I(x) dx_{i_1}\wedge \cdots\wedge dx_{i_k},$$
an exterior form with coefficients depending on the positions.

(We will be using sometimes simplifying notation $$\xi_i = dx_i$$ or $$\eta_i=dy_i$$.)

• Pullback maps: Let $$\Omega_1,\Omega_2$$ be two Euclidean domains
(not necessarily embedded in the same space) such that
$$\psi:\Omega_1 \to \Omega_2$$ is a smooth (continuously differentiable enough times) mapping between them.Let $$f$$ be a real function on $$\Omega_2$$. Then one can pull it back, by defining
$(\psi^*f)(x):=f(\psi(x)).$There is no mystery here – this is just a way to substitute.

Example: If $$\psi:\Real^2\to\Real^3$$ is given by
$\psi(x_1,x_2)=(x_1^2, x_1x_2, x_2^2),$
and $$f:\Real^3\to\Real$$ is a linear function $$ay_1+by_2+cy_3+d$$, then
$\psi^*f(x_1,x_2)=ax_1^2+bx_1x_2+cx_2^2+d.$

• Similarly one can define the pullback of a differential form. Again, the pullback of a differential form is just a substitution.

If $$\omega$$ is a $$k$$-form on $$\Omega_2$$, given by
$$\omega(x) = \sum_{I=\{i_1<i_2<\ldots<i_k\}} c_I(x) dx_{i_1}\wedge \cdots\wedge dx_{i_k},$$
and $$\psi:\Omega_1\to\Omega_2$$ is given, in coordinates, as
$x_i=\psi_i(y_1,\ldots,y_m),$
then
$(\psi^*\omega)(y)=\sum_{I=\{i_1<i_2<\ldots<i_k\}} c_I(\psi(y)) dx_{i_1}(y)\wedge \cdots\wedge dx_{i_k}(y)= \sum_{I=\{i_1<i_2<\ldots<i_k\}} c_I(\psi(y))(\sum_l\frac{\pd x_{i_1}}{\pd y_l}dy_l)(y)\wedge \cdots\wedge (\sum_l\frac{\pd x_{i_1}}{\pd y_l}(y)dy_l),$

Example: if $$\psi(r,\phi)=(r\cos(\phi),r\sin(\phi))=(x,y)$$,
then
$\psi^*(dx\wedge dy)=(\cos(\phi)dr-r\sin(\phi)d\phi)\wedge(\sin(\phi)dr+r\cos(\phi)d\phi)=r dr\wedge d\phi.$

• If $$\gamma$$ is a curve in
$$\Omega_1$$, it is mapped by $$\psi$$ into a curve on $$\Omega_2$$, again, by composition.
• One can verify immediately that for 1-forms
the composed path integrals of $$\psi^*\omega(\gamma)$$ and its image in $$\Omega_2$$
are the same:
$$\underbrace{\int_{\psi(\gamma)}df}_{\text{Integration in } \Omega_2} = \underbrace{\int_\gamma d \psi^*(f)}_{\text{Integration in } \Omega_1}.$$

• Examples
1. Define the 1-form $$\omega_1 = xdx + ydy = rdr$$ – this is, clearly, just the differential $$dr^2/2$$.
2. Another 1-form $$\omega_2 = xdy-ydx$$. Applied on a vector $$f=(s,t)$$ at a point $$r=(x,y)$$, it evaluates to
$$\omega_2(v) = \frac{xt – ys}.$$
The physical meaning is therefore the angular momentum of the force $$v$$ applied at $$r$$.
3. One more form: $$\omega_3=\omega_2/|r|^2$$ – defined everywhere outside of the origin.
In the polar coordinates $$d\phi := \frac{\mathbf{r}\times v}{|\mathbf{r}|^2}$$,
and we get the well-known identity:
$$\omega_1 \wedge \omega_2 = (rdr)\wedge d\phi = \frac{1}{2} (xdx+ydy)\wedge \frac{xdy+ydx}{x^2+y^2} = \frac{x^2dx\wedge dy – y^2 dy\wedge dx}{x^2+y^2} = dx\wedge dy$$