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Change of variables for differential forms

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Given $F \colon {\bf R} \rightarrow {\bf R}$, a smooth function: $x = F(x')$.

One thing we can observe is what happens under $F$ to the exterior derivative of the forms. For the $0$-forms: $$dx = dF(x') = F'(x')dx.$$

So, questions:

  • (a) What happens to $1$, $dx$, $x dx$, etc under $F$?
  • (b) Same question for $F \colon {\bf R}^2 \rightarrow {\bf R}^2$.
  • (c) Given $F \colon R_1 \rightarrow R_2$, what function does $F$ generate between $\Omega^k(R_1)$ and $\Omega_k(R_2)$?

Note: The problem is about change of variables.

Now, let's assume $R_1 = {\bf R}^2$ and $R_2 = {\bf R}^2$. Then $F \colon {\bf R}^2 \rightarrow {\bf R}^2$ is given by $$(x,y) = F(x',y') = (f(x',y'),g(x',y')).$$

A simple question: what is $dx = ?$

Consider $$x = f(x',y'),$$ $$dx = d(f(x',y')) = \frac{\partial f}{\partial x'} dx' + \frac{\partial f}{\partial y'} dy'.$$ And $dy$ behaves similarly.

Example: Suppose $F(x',y') = (xy', x'+y')$. Then $dx = y' dx' + dy'$. Observe that this is the differential in $R_1$ written in terms of that in $R_2$.

For question (c), let's consider $0$-forms only first, i.e., functions.

What does $F$ create between $\Omega^0(R_1)$ and $\Omega^0(R_2)$?

Recall:

  • $\Omega^0(R_1) = $ functions from $R_1$ to ${\bf R}$, $f_1$.
  • $\Omega^0(R_2) = $ functions from $R_2$ to ${\bf R}$, $f_2$.

We connect $f_1$ and $f_2$ in this commutative diagram:

Exam1Problem6.png

Question: How do we get $f_2$ from $f_1$? Or, how do we express $f_2$ in terms of $f_1$?

Hmm... the thing is, the relation between $f_1$ and $f_2$ and $F$ is via the compositions and we can compose only this way: $$f_2 \circ F!$$ The thing is $f_1 \circ F$ (or $F \circ f_2$ etc) just doesn't make sense.

So, $f_1$ is obtained from $f_2$ via composition with $F$, not vice versa as we expected!

Here $f_2$ is taken to $f_2 \circ F$ via a certain function: $$F^*(f_2)=f_2 \circ F,$$ or $$F^*:f_2 \longmapsto f_1=f_2 \circ F.$$ It is fully determined by $F$, hence the notation. This is where the new function operates: $$\Omega^0(R_2) \ni f_2 \stackrel{F^*}{\longmapsto} f_1 \in \Omega^0(R_1).$$ So, we have a function: $$F^*:\Omega^0(R_2) \rightarrow \Omega^0(R_1).$$

The arrow points in the opposite direction!

Indeed, compare: $$F \colon R_1 \rightarrow R_2$$ and $$F^* \colon \Omega^0(R_1) \leftarrow \Omega^0(R_2).$$

Same for forms of all degrees.

This is an instance of the general idea of covariance vs contravariance of functors, in category theory.

We take this idea one step further -- to de Rham cohomology. Our function $$F \colon R_1 \rightarrow R_2$$ generates a linear operator $$F^* \colon H^k_{dR}(R_1) \leftarrow H^k_{dR}(R_2).$$