\(\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**- Define the 1-form \(\omega_1 = xdx + ydy = rdr \) – this is, clearly, just the differential \(dr^2/2\).
- 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 \(r\). - 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 $$

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