Changes of variables

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

R,

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.

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Objectives

You should be able to:

Determine how one-forms transform under changes of variables on $R .$
State the transformation property of one-forms in terms of the mathematical notion of pullback.

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Subsection 2.3.1 How functions and one-forms on $R$ transform under changes of variables

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Consider a smooth function

f (x)

of a single variable

x,

i.e.

f : U \to R

for some open subset

U \subseteq R .

Now suppose that we think of

x

itself as a smooth function of another variable

t,

that is,

x = x (t) .

What happens to the function

f ?

Well, it's pretty simple: our function becomes

f (x (t)),

which defines a new function, which we could call

g (t) = f (x (t)) .

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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

R

that is exact, i.e. a differential of a function:

ω = d F .

We can write the one-form as

ω = d F (x) = \frac{d F}{d x} d x

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in terms of a real variable

x .

What happens if we think of

x

itself as a smooth function of another variable

t,

that is,

x = x (t) ?

We can use what we learned above about functions. We are interested in

d F (x (t)) .

If we define

G (t) = F (x (t))

as above, then our one-form can be written as

d G (t) = \frac{d G}{d t} d t .

But, using the chain rule of calculus, we know that

\frac{d G}{d t} = \frac{d}{d t} (F (x (t)) = \frac{d F}{d x} \frac{d x}{d t} .

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So by changing variable from

x

t,

we get a new one-form, let's call it

η,

defined by

η = \frac{d G}{d t} d t = (\frac{d F}{d x} \frac{d x}{d t}) d t .

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We now generalize this to all one-forms on

R

(or open subsets thereof), not just exact one-forms. Given a one-form

ω = f (x) d x

written in terms of a variable

x,

if we think of

x = x (t)

as a function of a new variable

t,

then by changing variable from

x

t

we get a new one-form

η :

η = (f (x (t)) \frac{d x}{d t}) d t .

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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.

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Remark 2.3.1.

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

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Example 2.3.2. An example of a change of variables.

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

η = f (x (t)) \frac{d x}{d t} d t = (\sin t)^{2} \cos t d t .

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Subsection 2.3.2 The pullback of functions and one-forms on $R$

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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 R

for some open subset

U \subseteq R .

We write

f (x)

in terms of a variable

x .

Now suppose that

x = ϕ (t),

for some smooth function

ϕ : V \to U,

where

V \subseteq R

is also an open subset of

R .

As described above, changing variables from

x

t

amounts to defining a new function

g (t) = f (ϕ (t)) .

This new function is simply the composition of

f

and

ϕ :

g = f \circ ϕ : V \to R .

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We call this new function “the pullback of

f

”.

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Definition 2.3.3. The pullback of a function on $R$ .

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

ϕ^{*} f := f \circ ϕ : V \to R .

Explicitly, the pullback can be written as $ϕ^{*} f (t) = f (ϕ (t)) .$

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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{ϕ}{\to} U \overset{f}{\to} R,

while our original function was from

U

R,

by composition we “pull it back” to a function from

V

R .

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We can define a similar concept for one-forms on

R,

using the discussion above about how they transform under changes of variables.

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Definition 2.3.4. The pullback of a one-form on $R$ .

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

ϕ^{*} ω = (f (ϕ (t)) \frac{d ϕ}{d t}) d t = (ϕ^{*} f (t) \frac{d ϕ}{d t}) d t

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Note that when calculating the pullback of a one-form, it is very important not to forget the

\frac{d ϕ}{d t}

term!

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Example 2.3.5. Change of variables as pullback.

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

ϕ^{*} ω = f (ϕ (t)) \frac{d ϕ}{d t} d t = (\sin t)^{2} \cos t d t .

This is of course the same thing as implementing the change of variables $x \to t$ in the one-form $ω .$

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Exercises 2.3.3 Exercises

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1.

Consider the one-form $ω = e^{x} \sin (x) d x$ on $R,$ and the smooth function $ϕ : R_{> 0} \to R$ with $ϕ (t) = \ln (t)$ (where $R_{> 0}$ is the set of positive real numbers). Find the pullback one-form $ϕ^{*} ω .$ Where is the one-form $ϕ^{*} ω$ defined?

Solution.

First, since $ϕ : R_{> 0} \to R,$ and $ω$ is a one-form defined on all of $R,$ by definition of the pullback we see that the pullback one-form $ϕ^{*} ω$ is defined only on $R_{> 0},$ i.e. for all positive real numbers. Using the definition of pullback, we find its expression as:

\begin{aligned} ϕ^{*} ω = & e^{ϕ (t)} \sin (ϕ (t)) \frac{d ϕ}{d t} d t \\ = & e^{\ln (t)} \sin (\ln (t)) \frac{1}{t} d t \\ = & t \sin (\ln (t)) \frac{1}{t} d t \\ = & \sin (\ln (t)) d t . \end{aligned}

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2.

Consider the one-form $ω = \frac{1}{x} d x$ defined on $R_{> 0},$ and the smooth function $ϕ : R \to R_{> 0}$ defined by $ϕ (t) = e^{t} .$ Find the pullback one-form $ϕ^{*} ω .$

Solution.

First, by definition of the pullback we see that $ϕ^{*} ω$ is defined on all of $R .$ We find its expression to be:

\begin{aligned} ϕ^{*} ω = & \frac{1}{ϕ (t)} ϕ^{'} (t) d t \\ = & e^{- t} e^{t} d t \\ = & d t . \end{aligned}

How simple! :-)

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

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3.

Following up on the previous exercise, show that for an exact one-form $ω = d f$ on $U \subseteq R,$ and a smooth function $ϕ : V \to U,$ the following property of the pullback is satisfied:

ϕ^{*} (d f) = d (ϕ^{*} f) .

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:

d (ϕ^{*} f) = d (f (ϕ (t)) = \frac{d f (ϕ (t))}{d t} d t .

Using the chain rule, this can be written as

\begin{aligned} d (ϕ^{*} f) = & f^{'} (ϕ (t)) \frac{d ϕ}{d t} d t \\ = & ϕ^{*} (f^{'} (x) d x) \\ = & ϕ^{*} (d f), \end{aligned}

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

Section 2.3 Changes of variables

Objectives

Subsection 2.3.1 How functions and one-forms on R transform under changes of variables

Remark 2.3.1.

Example 2.3.2. An example of a change of variables.

Subsection 2.3.2 The pullback of functions and one-forms on R

Definition 2.3.3. The pullback of a function on R.

Definition 2.3.4. The pullback of a one-form on R.

Example 2.3.5. Change of variables as pullback.

Exercises 2.3.3 Exercises

1.

2.

3.

Subsection 2.3.1 How functions and one-forms on $R$ transform under changes of variables

Subsection 2.3.2 The pullback of functions and one-forms on $R$

Definition 2.3.3. The pullback of a function on $R$ .

Definition 2.3.4. The pullback of a one-form on $R$ .