The pullback of a one-form

Section 2.4 The pullback of a one-form

All right, time to get serious! :-) In the previous section we introduced the notion of the pullback of a one-form with respect to a function

ϕ : V \to U

with

U, V \subseteq R

open subsets. But this notion of pullback can be generalized, and will become essential to develop our theory of integration (in fact, perhaps this class should be called “the power of the pullback”!). In this section we provide a more general definition of the pullback of a one-form.

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Objectives

You should be able to:

Determine the pullback of a one-form in general.
State and use the three fundamental properties of the pullback of one-forms.

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Let

ω

be a one-form on

U \subseteq R^{n},

where

n

is a positive integer. We now consider a smooth function

ϕ : V \to U,

where

V \subseteq R^{m},

with again

m

a positive integer. Note that

m

and

n

don't have to be the same: we could have, say

U \subseteq R^{3},

and

V \subseteq R^{2} .

Our goal is to define the pullback

ϕ^{*} ω,

which should be a one-form on

V .

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Just to be concrete: we could take, for instance, a one-form

ω

R^{3},

and a smooth function

ϕ : R^{2} \to R^{3} .

The pullback

ϕ^{*} ω

should then be a one-form on

R^{2} .

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Note that our notion of pullback should generalize the definition of pullback in Definition 2.3.4, which should consist in the case with

m = n = 1 .

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Subsection 2.4.1 The pullback of a function

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Let us first define the pullback of a function in this context, generalizing Definition 2.3.3.

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Definition 2.4.1. The pullback of a function.

Let $U \subseteq R^{n}$ and $V \subseteq R^{m}$ be open subsets, where $m, n$ are positive integers. Let $f : U \to R$ 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 .

Explicitely, if we write $t \in V$ for an $m$ -dimensional vector in $V,$ then

ϕ^{*} f (t) = f (ϕ (t)),

where $ϕ (t)$ is an $n$ -dimensional vector in $U .$

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Example 2.4.2. The pullback of a function from $R^{3}$ to $R$ .

To make things more concrete, let us look at a specific example. Suppose that $f$ is a smooth function on $R^{3},$ that is $f : R^{3} \to R .$ Let $ϕ : R \to R^{3}$ be another smooth function, which takes a point in $R$ and maps it to a vector in $R^{3}$ (so it is a vector-valued function). We can write $f$ explicitly as $f = f (x, y, z) .$ As for the vector-valued function $ϕ,$ we write $ϕ (t) = (x (t), y (t), z (t)) .$ Then the pullback $ϕ^{*} f : R \to R$ is simply the composition:

ϕ^{*} f (t) = f (ϕ (t)) = f (x (t), y (t), z (t)) .

For instance, if $f (x, y, z) = x y + z,$ and $ϕ (t) = (t, t^{2}, 1),$ then

ϕ^{*} f (t) = f (t, t^{2}, 1) = t^{3} + 1.

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Example 2.4.3. The pullback of a function from $R^{3}$ to $R^{2}$ .

We can do the same thing but pulling back to $R^{2}$ instead of $R .$ Suppose that $f$ is a smooth function on $R^{3},$ that is $f : R^{3} \to R .$ Let $ϕ : R^{2} \to R^{3}$ be another smooth function, which takes a point in $R^{2}$ and maps it to a vector in $R^{3} .$ We can write $f$ explicitly as $f = f (x, y, z) .$ As for the vector-valued function $ϕ,$ we write $ϕ (t_{1}, t_{2}) = (x (t_{1}, t_{2}), y (t_{1}, t_{2}), z (t_{1}, t_{2})) .$ Then the pullback $ϕ^{*} f : R \to R$ is simply the composition:

ϕ^{*} f (t_{1} t_{2}) = f (ϕ (t_{1}, t_{2})) = f (x (t_{1}, t_{2}), y (t_{1}, t_{2}), z (t_{1}, t_{2})) .

For instance, if $f (x, y, z) = x y + z,$ and $ϕ (t_{1}, t_{2}) = (t_{1}, t_{2}^{2}, t_{1} + t_{2}),$ then

ϕ^{*} f (t_{1}, t_{2}) = f (t_{1}, t_{2}^{2}, t_{1} + t_{2}) = t_{1} t_{2}^{2} + t_{1} + t_{2} .

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Subsection 2.4.2 An axiomatic definition of the pullback of a one-form

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We will take an axiomatic approach to the definition of the pullback of a one-form. Let us first recall three important properties of one-forms (from Subsection 2.1.1 and Definition 2.2.5):

If $ω$ and $η$ are one-forms on $U,$ then $ω + η$ is a one-form on $U .$
If $ω$ is a one-form on $U$ and $f$ a smooth function on $U,$ then $f ω$ is a one-form on $U .$
An exact one-form is a one-form $ω$ that can be written as the differential of a function $f$ on $U :$ $ω = d f .$

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We now want the pullback to be consistent with these properties. More precisely, we require that the pullback

ϕ^{*}

satisfies the following properties:

$ϕ^{*} (ω + η) = ϕ^{*} ω + ϕ^{*} η .$
$ϕ^{*} (f ω) = (ϕ^{*} f) (ϕ^{*} ω) .$
$ϕ^{*} (d f) = d (ϕ^{*} f) .$

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It turns out that this is completely sufficient to fully determine the pullback of any one-form. Let us see why.

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Lemma 2.4.4. The pullback of $d x$ .

Let $d x$ be the basic one-form on $R^{3} .$ Let $ϕ : V \to R^{3}$ be a smooth function, where $V \subseteq R^{m}$ is an open subset, with $m$ a positive integer. Write $t = (t_{1}, \dots, t_{m})$ for an $m$ -dimensional vector in $V,$ and $ϕ (t) = (x (t), y (t), z (t)) .$ Then

ϕ^{*} (d x) = \sum_{i = 1}^{m} \frac{\partial x}{\partial t_{i}} d t_{i} .

As an example, if $m = 3,$ we would get

ϕ^{*} (d x) = \frac{\partial x}{\partial t_{1}} d t_{1} + \frac{\partial x}{\partial t_{2}} d t_{2} + \frac{\partial x}{\partial t_{3}} d t_{3} .

The same formula for $ϕ^{*} (d x)$ remains true if $d x$ is considered to be a one-form on $R^{2},$ $R,$ or more generally $R^{n},$ instead of $R^{3} .$

We also note that the same result holds for the basic one-forms $d y$ and $d z,$ with $x$ replaced by $y$ and $z$ respectively.

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Proof.

This follows from the third axiomatic property that we are imposing on the pullback. Recall from Remark 2.2.4 that we can think of $d x$ as the differential $d f$ of the function $f (x, y, z) = x .$ By the third property of pullbacks, we want to impose that

ϕ^{*} (d x) = ϕ^{*} (d f) = d (ϕ^{*} f) .

From Definition 2.4.1, we can calculate $ϕ^{*} f .$ We get $ϕ^{*} f (t) = x (t) .$ We thus obtain

ϕ^{*} (d x) = d x (t) = \sum_{i = 1}^{m} \frac{\partial x}{\partial t_{i}} d t_{i},

where we use the definition of the differential of the function $x (t) .$

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This result enables us to write down a general formula for the pullback of a one-form. For clarity, we will only write it down for a one-form on

R^{3},

but it is clear what the similar result should be for a one-form in

R^{2}

R .

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Lemma 2.4.5. The pullback of a one-form.

Let $ω = f d x + g d y + h d z$ be a one-form on an open subset $U \subseteq R^{3} .$ Let $ϕ : V \to U$ be a smooth function, where $V \subseteq R^{m}$ is an open subset, with $m$ a positive integer. Write $t = (t_{1}, \dots, t_{m})$ for an $m$ -dimensional vector in $V,$ and $ϕ (t) = (x (t), y (t), z (t)) .$ Then

ϕ^{*} ω = f (ϕ (t)) \sum_{i = 1}^{m} \frac{\partial x}{\partial t_{i}} d t_{i} + g (ϕ (t)) \sum_{i = 1}^{m} \frac{\partial y}{\partial t_{i}} d t_{i} + h (ϕ (t)) \sum_{i = 1}^{m} \frac{\partial z}{\partial t_{i}} d t_{i} .

A similar formula holds if we start with a one-form on $R^{n}$ instead of $R^{3} .$

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Proof.

To prove this result, we use Lemma 2.4.4 (and the similar result for $d y$ and $d z$ ), and the first and second axiomatic properties that we are imposing on the pullback. Using the first and second properties, we can write:

ϕ^{*} ω = ϕ^{*} (f d x + g d y + h d z) = (ϕ^{*} f) ϕ^{*} (d x) + (ϕ^{*} g) ϕ^{*} (d y) + (ϕ^{*} h) ϕ^{*} (d z) .

We then use Lemma 2.4.4 to evaluate $ϕ^{*} (d x), ϕ^{*} (d y)$ and $ϕ^{*} (d z),$ and from Definition 2.4.1 we know that $ϕ^{*} f (t) = f (ϕ (t)),$ and similarly for $g$ and $h .$

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Example 2.4.6. The pullback of a one-form from $R^{3}$ to $R$ .

Suppose that $ω = f d x + g d y + h d z$ is a one-form on $R^{3} .$ Let $ϕ : R \to R^{3}$ be a smooth function, which takes a point in $R$ and maps it to a vector in $R^{3} .$ We write $ϕ (t) = (x (t), y (t), z (t)) .$ Then the pullback $ϕ^{*} ω$ is a one-form on $R$ given by:

ϕ^{*} ω = (f (ϕ (t)) \frac{d x}{d t} + g (ϕ (t) \frac{d y}{d t} + h (ϕ (t)) \frac{d z}{d t}) d t .

For instance, if $ω = x d x + x y d y + z^{2} d z,$ and $ϕ (t) = (x (t), y (t), z (t)) = (t^{2}, t, 1),$ then

\begin{aligned} ϕ^{*} ω = & (x (t) \frac{d x}{d t} + x (t) y (t) \frac{d y}{d t} + z (t)^{2} \frac{d z}{d t}) d t \\ = & ((t^{2}) (2 t) + (t^{2}) (t) (1) + (1) (0)) d t \\ = & 3 t^{3} d t . \end{aligned}

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Example 2.4.7. The pullback of a one-form from $R^{3}$ to $R^{2}$ .

Suppose that $ω = f d x + g d y + h d z$ is a one-form on $R^{3} .$ Let $ϕ : R^{2} \to R^{3}$ be a smooth function, which takes a point in $R^{2}$ and maps it to a vector in $R^{3} .$ We write $ϕ (t) = (x (t), y (t), z (t)),$ with $t = (t_{1}, t_{2}) .$ Then the pullback $ϕ^{*} ω$ is a one-form on $R^{2}$ given by:

\begin{aligned} ϕ^{*} ω = & f (ϕ (t)) (\frac{\partial x}{\partial t_{1}} d t_{1} + \frac{\partial x}{\partial t_{2}} d t_{2}) + g (ϕ (t)) (\frac{\partial y}{\partial t_{1}} d t_{1} + \frac{\partial y}{\partial t_{2}} d t_{2}) \\ + h (ϕ (t)) (\frac{\partial z}{\partial t_{1}} d t_{1} + \frac{\partial z}{\partial t_{2}} d t_{2}) . \end{aligned}

For instance, if $ω = x d x + x y d y + z^{2} d z,$ and $ϕ (t) = (x (t), y (t), z (t)) = (t_{1} t_{2}, t_{2}, t_{1} + t_{2}),$ then

\begin{aligned} ϕ^{*} ω = & x (ϕ (t)) (\frac{\partial x}{\partial t_{1}} d t_{1} + \frac{\partial x}{\partial t_{2}} d t_{2}) + x (ϕ (t)) y (ϕ (t)) (\frac{\partial y}{\partial t_{1}} d t_{1} + \frac{\partial y}{\partial t_{2}} d t_{2}) \\ + z (ϕ (t))^{2} (\frac{\partial z}{\partial t_{1}} d t_{1} + \frac{\partial z}{\partial t_{2}} d t_{2}) . \\ = & (t_{1} t_{2}) (t_{2} d t_{1} + t_{1} d t_{2}) + (t_{1} t_{2}) (t_{2}) (d t_{2}) + (t_{1} + t_{2})^{2} (d t_{1} + d t_{2}) \\ = & (t_{1} t_{2}^{2} + (t_{1} + t_{2})^{2}) d t_{1} + (t_{1}^{2} t_{2} + t_{1} t_{2}^{2} + (t_{1} + t_{2})^{2}) d t_{2} . \end{aligned}

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Example 2.4.8. Consistency check: the pullback of a one-form from $R$ to $R$ .

As a consistency check, we show that the pullback of a one-from from $R$ to $R$ reduces to Definition 2.3.4. Let $ω = f (x) d x$ on $U \subseteq R,$ and $ϕ : V \to U$ with $V \subseteq R .$ We write $ϕ (t) = x (t) .$ Then the pullback $ϕ^{*} ω$ is the one-form on $V$ given by:

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

which indeeds reproduces Definition 2.3.4 with our notation $ϕ (t) = x (t) .$

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We now have a very general definition of the pullback of a one-form. This will turn out to be very useful to define the integral of a one-form, which is what we now turn to.

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Exercises 2.4.3 Exercises

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1.

Consider the function $f : R^{2} \to R$ given by $f (x, y) = e^{x + y} + x + y,$ and the function $ϕ : R^{3} \to R^{2}$ given by $ϕ (u, v, w) = (u + v, v + w) .$ Find the pullback $ϕ^{*} f .$ What is its domain?

Solution.

First, as $f : R^{2} \to R$ and $ϕ : R^{3} \to R^{2},$ we see that the composition $ϕ^{*} f = f \circ ϕ : R^{3} \to R^{2} \to R,$ i.e. the pullback $ϕ^{*} f$ is a function from $R^{3}$ to $R .$ So its domain is $R^{3} .$

We calculate its expression by composition:

\begin{aligned} ϕ^{*} f (u, v, w) = & f (u + v, v + w) \\ = & e^{(u + v) + (v + w)} + (u + v) + (v + w) \\ = & e^{u + 2 v + w} + u + 2 v + w . \end{aligned}

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2.

Consider the one-form $ω = x^{2} d x$ on $R,$ and the function $ϕ : R^{2} \to R$ given by $ϕ (u, v) = u,$ which projects on the $u$ -axis. Find the pullback one-form $ϕ^{*} ω$ on $R^{2} .$ Interpret the result.

Solution.

Let us write $ω = f d x = x^{2} d x .$ By the definition of pullback, we get:

\begin{aligned} ϕ^{*} ω = & f (ϕ (u, v)) (\frac{\partial ϕ}{\partial u} d u + \frac{\partial ϕ}{\partial v} d v) \\ = & u^{2} (1 d u + 0 d v) \\ = & u^{2} d u . \end{aligned}

We see that the pullback one-form looks the same, but written in terms of $u$ instead of $x .$ However, $ϕ^{*} ω$ is defined on $R^{2},$ while $ω$ was defined on $R .$ Since the function $ϕ$ here simply projects on the $u$ -axis, what the pullback does here is extend the one-form uniformly in the $v$ -coordinate on the $u v$ -plane; at any two points $(u, v_{1})$ and $(u, v_{2}),$ the one-form will be the same. Conceptually, this is what happens when we pullback using a “forgetful map”, i.e. a map that somehow “forgets” some information (in this case, the $v$ -coordinate). The pullback then extends the object uniformly across the forgotten structure.

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3.

Consider the one-form $ω = x^{2} d x + y^{2} d y$ on $R^{2},$ and the map $Φ : R^{2} \to R^{2}$ with $Φ (r, θ) = (r \cos θ, r \sin θ),$ which defines polar coordinates. Find the pullback $Φ^{*} ω .$

Solution.

We write $ω = f d x + g d y = x^{2} d x + y^{2} d y,$ and $Φ (r, θ) = (x (r, θ), y (r, θ)) .$ Then:

\begin{aligned} Φ^{*} ω = & f (Φ (r, θ)) (\frac{\partial x}{\partial r} d r + \frac{\partial x}{\partial θ} d θ) + g (Φ (r, θ)) (\frac{\partial y}{\partial r} d r + \frac{\partial y}{\partial θ} d θ) \\ = & r^{2} \cos^{2} θ (\cos θ d r - r \sin θ d θ) + r^{2} \sin^{2} θ (\sin θ d r + r \cos θ d θ) \\ = & r^{2} (\cos^{3} θ + \sin^{3} θ) d r + r^{3} (\sin^{2} θ \cos θ - \cos^{2} θ \sin θ) d θ . \end{aligned}

The notion of pullback allows us to easily calculate how one-forms change under changes of coordinates, such as going from Cartesian to polar coordinates in this case.

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4.

Consider the one-form $ω = \frac{z^{2} d z}{\sqrt{x^{2} + y^{2}}}$ on $U = {(x, y, z) \in R^{3} | (x, y, z) \neq (0, 0, z)}$ (this is $R^{3}$ with the $z$ -axis removed), and the function $ϕ : V \to U$ with $ϕ (r, θ, ζ) = r (\cos θ, \sin θ, ζ),$ and $V = {(r, θ, ζ) \in R^{3} | r \neq 0} .$ Determine the pullback one-form $ϕ^{*} ω .$

Solution.

By definition of the pullback, we get:

\begin{aligned} ϕ^{*} ω = & \frac{1}{r} r^{2} ζ^{2} (\frac{\partial}{\partial r} (r ζ) d r + \frac{\partial}{\partial θ} (r ζ) d θ + \frac{\partial}{\partial ζ} (r ζ) d ζ) \\ = & r ζ^{2} (ζ d r + r d ζ) . \end{aligned}

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5.

Let $ω$ be a one-form on $R^{3},$ and $I d : R^{3} \to R^{3}$ the identity function defined by $I d (x, y, z) = (x, y, z) .$ Show that $I d^{*} ω = ω .$

Solution.

Write $ω = f d x + g d y + h d z .$ By definition of the pullback, we get:

\begin{aligned} I d^{*} ω = & f (x, y, z) (\frac{\partial}{\partial x} (x) d x + \frac{\partial}{\partial y} (x) d y + \frac{\partial}{\partial z} (x) d z) \\ + g (x, y, z) (\frac{\partial}{\partial x} (y) d x + \frac{\partial}{\partial y} (y) d y + \frac{\partial}{\partial z} (y) d z) \\ + h (x, y, z) (\frac{\partial}{\partial x} (z) d x + \frac{\partial}{\partial y} (z) d y + \frac{\partial}{\partial z} (z) d z) \\ = & f (x, y, z) d x + g (x, y, z) d y + h (x, y, z) d z \\ = & ω, \end{aligned}

which completes the proof.

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6.

Let $ω$ be a one-form $U \subseteq R,$ and $ϕ : V \to U$ and $α : W \to V$ be smooth functions, with $V, W \subseteq R$ open subsets. Show that

(ϕ \circ α)^{*} ω = α^{*} (ϕ^{*} ω) .

In other words, it doesn't matter whether we pullback in one or two steps through the chain of maps $W \overset{α}{\to} V \overset{ϕ}{\to} U .$

We note here that while the exercise is only asking you to prove it for open subsets $U, V, W \subseteq R,$ this property is true in general, not just in $R .$

Solution.

Let us write $ω = f (x) d x,$ $ϕ = ϕ (t),$ and $α = α (u) .$ On the one hand, we have:

(ϕ \circ α)^{*} ω = f (ϕ (α (u))) \frac{d}{d u} (ϕ (α (u))) d u .

On the other hand, we have

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

and

α^{*} (ϕ^{*} ω) = f (ϕ (α (u)) (\frac{d}{d t} ϕ (t)) |_{t = α (u)} \frac{d}{d u} α (u) d u .

But

\frac{d}{d u} (ϕ (α (u))) = (\frac{d}{d t} ϕ (t)) |_{t = α (u)} \frac{d}{d u} α (u)

by the chain rule, and hence $(ϕ \circ α)^{*} ω = α^{*} (ϕ^{*} ω) .$

Section 2.4 The pullback of a one-form

Objectives

Subsection 2.4.1 The pullback of a function

Definition 2.4.1. The pullback of a function.

Example 2.4.2. The pullback of a function from R3 to R.

Example 2.4.3. The pullback of a function from R3 to R2.

Subsection 2.4.2 An axiomatic definition of the pullback of a one-form

Lemma 2.4.4. The pullback of dx.

Proof.

Lemma 2.4.5. The pullback of a one-form.

Proof.

Example 2.4.6. The pullback of a one-form from R3 to R.

Example 2.4.7. The pullback of a one-form from R3 to R2.

Example 2.4.8. Consistency check: the pullback of a one-form from R to R.

Exercises 2.4.3 Exercises

1.

2.

3.

4.

5.

6.

Example 2.4.2. The pullback of a function from $R^{3}$ to $R$ .

Example 2.4.3. The pullback of a function from $R^{3}$ to $R^{2}$ .

Lemma 2.4.4. The pullback of $d x$ .

Example 2.4.6. The pullback of a one-form from $R^{3}$ to $R$ .

Example 2.4.7. The pullback of a one-form from $R^{3}$ to $R^{2}$ .

Example 2.4.8. Consistency check: the pullback of a one-form from $R$ to $R$ .