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Section 2.3 Changes of variables

Recall from the introduction that our goal is to construct a theory of integration over curves, surfaces, and volumes. Our guiding light is that we want the theory to be “reparametrization-invariant” and “oriented”. In the last few subsections, we introduced the concept of one-forms and their associated vector fields. In the next section, we will see that one-forms become the objects that will be integrated over curves. Thus, to study reparametrization-invariance, we need to know how one-forms transform under changes of variables. This is what we study in this section. For the time being, we focuse only on functions and one-forms on \(\mathbb{R}\text{,}\) and remain informal and hand-wavy; our goal is to get a feeling for how one-forms transform. Studying how one-forms transform will encourage us to introduce a little bit of mathematical formalism: the concept of “pullbacks”. We will generalize further the concept of pullbacks in the next section.

Subsection 2.3.1 How functions and one-forms on \(\mathbb{R}\) transform under changes of variables

Consider a smooth function \(f(x)\) of a single variable \(x\text{,}\) i.e. \(f: U \to \mathbb{R}\) for some open subset \(U \subseteq \mathbb{R}\text{.}\) Now suppose that we think of \(x\) itself as a smooth function of another variable \(t\text{,}\) that is, \(x = x(t)\text{.}\) What happens to the function \(f\text{?}\) Well, it's pretty simple: our function becomes \(f(x(t))\text{,}\) which defines a new function, which we could call \(g(t) = f(x(t))\text{.}\)

What about one-forms? How do these transform under changes of variables? To see what is going on, let us first consider a one-form on \(\mathbb{R}\) that is exact, i.e. a differential of a function: \(\omega = d F\text{.}\) We can write the one-form as

\begin{equation*} \omega = d F(x) = \frac{dF}{dx}\ dx \end{equation*}

in terms of a real variable \(x\text{.}\) What happens if we think of \(x\) itself as a smooth function of another variable \(t\text{,}\) that is, \(x = x(t)\text{?}\) We can use what we learned above about functions. We are interested in \(d F(x(t))\text{.}\) If we define \(G(t) = F(x(t))\) as above, then our one-form can be written as \(d G(t) = \frac{dG}{dt}\ dt\text{.}\) But, using the chain rule of calculus, we know that

\begin{equation*} \frac{dG}{dt} = \frac{d}{dt} \left( F(x(t) \right) = \frac{dF}{dx} \frac{dx}{dt}. \end{equation*}

So by changing variable from \(x\) to \(t\text{,}\) we get a new one-form, let's call it \(\eta\text{,}\) defined by

\begin{equation*} \eta = \frac{dG}{dt}\ dt = \left( \frac{dF}{dx} \frac{dx}{dt} \right)\ dt. \end{equation*}

We now generalize this to all one-forms on \(\mathbb{R}\) (or open subsets thereof), not just exact one-forms. Given a one-form \(\omega = f(x)\ dx\) written in terms of a variable \(x\text{,}\) if we think of \(x = x(t)\) as a function of a new variable \(t\text{,}\) then by changing variable from \(x\) to \(t\) we get a new one-form \(\eta\text{:}\)

\begin{equation*} \eta = \left(f(x(t)) \frac{dx}{dt}\right)\ dt. \end{equation*}

This defines how one-forms transform under changes of variables. As we will see, this transformation property is what lies behind the substitution formula for definite integrals.

Remark 2.3.1.

The upshot of this brief discussion is that it is easy to remember how one-forms in \(\mathbb{R}\) transform under changes of variables. If we write \(\omega = f(x)\ dx\text{,}\) and do a change of variable \(x=x(t)\text{,}\) then all we need to do is rewrite the coefficient function as a function of \(t\) by composition \(f(x(t))\text{,}\) and then “transform the differential” \(dx\) as \(dx(t) = \frac{dx}{dt}\ dt\text{.}\) This gives us the new one-form \(\eta = f(x(t)) \frac{dx}{dt}\ dt.\)

Consider the one-form \(\omega = x^2 dx = f(x)\ dx\) on \(\mathbb{R}\text{.}\) Let us do the change of variables \(x = \sin t\text{,}\) with \(\frac{dx}{dt} = \cos t\text{.}\) By changing variables from \(x\) to \(t\) we a get a new one-form

\begin{equation*} \eta = f(x(t)) \frac{dx}{dt}\ dt = (\sin t)^2 \cos t \ dt. \end{equation*}

Subsection 2.3.2 The pullback of functions and one-forms on \(\mathbb{R}\)

Our brief discussion above can be formalized mathematically in terms of the concept of “pullback”. Let us start with functions again, and be a little more formal. Given a function \(f: U \to \mathbb{R}\) for some open subset \(U \subseteq \mathbb{R}\text{.}\) We write \(f(x)\) in terms of a variable \(x\text{.}\) Now suppose that \(x = \phi(t)\text{,}\) for some smooth function \(\phi: V \to U\text{,}\) where \(V \subseteq \mathbb{R}\) is also an open subset of \(\mathbb{R}\text{.}\) As described above, changing variables from \(x\) to \(t\) amounts to defining a new function \(g(t) = f(\phi(t))\text{.}\) This new function is simply the composition of \(f\) and \(\phi\text{:}\)

\begin{equation*} g = f \circ \phi: V \to \mathbb{R}. \end{equation*}

We call this new function “the pullback of \(f\)”.

Definition 2.3.3. The pullback of a function on \(\mathbb{R}\).

Let \(U\subseteq \mathbb{R}\) and \(V \subseteq \mathbb{R}\) be open subsets, and \(f:U \to \mathbb{R}\) and \(\phi:V \to U\) be smooth functions. The pullback of \(f\), which is denoted by \(\phi^* f\text{,}\) is the smooth function

\begin{equation*} \phi^* f := f \circ \phi : V \to \mathbb{R}. \end{equation*}

Explicitly, the pullback can be written as \(\phi^* f(t) = f(\phi(t))\text{.}\)

In other words, the pullback of a function by another function just means that we are composing functions. It is called “pullback” because if we think of the chain of maps: \(V \overset{\phi}{\to} U \overset{f}{\to} \mathbb{R}\text{,}\) while our original function was from \(U\) to \(\mathbb{R}\text{,}\) by composition we “pull it back” to a function from \(V\) to \(\mathbb{R}\text{.}\)

We can define a similar concept for one-forms on \(\mathbb{R}\text{,}\) using the discussion above about how they transform under changes of variables.

Definition 2.3.4. The pullback of a one-form on \(\mathbb{R}\).

Let \(U\subseteq \mathbb{R}\) and \(V \subseteq \mathbb{R}\) be open subsets, \(\omega=f(x)\ dx\) be a one-form on \(U\text{,}\) and \(\phi:V \to U\) be a smooth function. The pullback of \(\omega\), which is denoted by \(\phi^* \omega\text{,}\) is the one-form on \(V\) defined by

\begin{equation*} \phi^* \omega = \left( f(\phi(t)) \frac{d \phi}{d t}\right)\ dt = \left( \phi^* f(t) \frac{d \phi}{d t}\right)\ dt \end{equation*}

Note that when calculating the pullback of a one-form, it is very important not to forget the \(\frac{d\phi}{d t}\) term!

Going back to Example 2.3.2, we could rephrase it as follows. We have a one-form \(\omega = x^2 dx = f(x)\ dx\) on \(\mathbb{R}\text{,}\) and a function \(\phi: \mathbb{R} \to \mathbb{R}\) given by \(\phi(t) = \sin t\text{.}\) The pullback one-form \(\phi^* \omega\) is then given by

\begin{equation*} \phi^* \omega = f(\phi(t)) \frac{d\phi}{dt}\ dt = (\sin t)^2 \cos t \ dt. \end{equation*}

This is of course the same thing as implementing the change of variables \(x \to t\) in the one-form \(\omega\text{.}\)

Exercises 2.3.3 Exercises

1.

Consider the one-form \(\omega = e^x \sin(x)\ dx\) on \(\mathbb{R}\text{,}\) and the smooth function \(\phi: \mathbb{R}_{>0} \to \mathbb{R}\) with \(\phi(t) = \ln(t)\) (where \(\mathbb{R}_{>0}\) is the set of positive real numbers). Find the pullback one-form \(\phi^* \omega\text{.}\) Where is the one-form \(\phi^* \omega\) defined?

Solution.

First, since \(\phi: \mathbb{R}_{>0} \to \mathbb{R}\text{,}\) and \(\omega\) is a one-form defined on all of \(\mathbb{R}\text{,}\) by definition of the pullback we see that the pullback one-form \(\phi^* \omega\) is defined only on \(\mathbb{R}_{>0}\text{,}\) i.e. for all positive real numbers. Using the definition of pullback, we find its expression as:

\begin{align*} \phi^* \omega =\amp e^{\phi(t)} \sin(\phi(t)) \frac{d \phi}{d t}\ dt\\ =\amp e^{\ln(t)} \sin(\ln(t)) \frac{1}{t}\ dt\\ =\amp t \sin(\ln(t)) \frac{1}{t}\ dt\\ =\amp \sin(\ln(t))\ dt. \end{align*}

2.

Consider the one-form \(\omega = \frac{1}{x}\ dx\) defined on \(\mathbb{R}_{> 0}\text{,}\) and the smooth function \(\phi: \mathbb{R} \to \mathbb{R}_{> 0}\) defined by \(\phi(t) = e^t\text{.}\) Find the pullback one-form \(\phi^* \omega\text{.}\)

Solution.

First, by definition of the pullback we see that \(\phi^* \omega\) is defined on all of \(\mathbb{R}\text{.}\) We find its expression to be:

\begin{align*} \phi^* \omega =\amp \frac{1}{\phi(t)} \phi'(t)\ dt \\ =\amp e^{-t} e^t\ dt\\ =\amp dt. \end{align*}

How simple! :-)

This is not really a surprise, since \(\omega = d \ln(x)\text{.}\) One property of the pullback of one-forms is that for exact one-forms, \(\phi^* d f = d(\phi^* f)\text{.}\) For \(f =\ln(x)\text{,}\) the pullback of the function is simply \(\phi^* f(t) = f(\phi(t)) = \ln(e^t) = t\text{,}\) and hence \(\phi^* d f = d \phi^* f = d t\text{,}\) as we found.

3.

Following up on the previous exercise, show that for an exact one-form \(\omega = df\) on \(U \subseteq \mathbb{R}\text{,}\) and a smooth function \(\phi: V \to U\text{,}\) the following property of the pullback is satisfied:

\begin{equation*} \phi^* (d f) = d(\phi^* f). \end{equation*}

In other words, the pullback commutes with the exterior derivative of a function.

Solution.

From the definition of the pullback of a function, we can write the right-hand-side as:

\begin{equation*} d(\phi^* f) = d(f(\phi(t)) = \frac{d f(\phi(t))}{d t}\ dt. \end{equation*}

Using the chain rule, this can be written as

\begin{align*} d( \phi^* f) =\amp f'(\phi(t)) \frac{d \phi}{d t} \ dt\\ =\amp \phi^* (f'(x)\ dx) \\ = \amp \phi^* (d f), \end{align*}

where we used the definition of the pullback of a one-form. This concludes the proof.