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Chapter 5: Integration

We know how to integrate real-valued functions of a real variable. We want to extend this idea to manifolds, in such a way that the integral is independent of the coordinate system used to compute it.

The integral of a real-valued function of a real variable is the limit of a sum of products of the values of the function on subintervals and the lengths of the increments of the independent variable in those subintervals:

\begin{equation} ∫_a^b{f} = ∫_a^b{f(x)dx} = lim_{Δ x_i → 0} ∑_i{f(x_i)} Δ x_i. \end{equation}

If we change variables $(x = g(y))$, then the form of the integral changes:

\begin{equation} \begin{aligned} ∫_a^b{f} &= ∫_a^b{f(x)dx}
&= ∫_g^{-1(a)}^{g^{-1}(b)}{f(g(y)) Dg(y)dy} \ &= ∫_g^{-1(a)}^{g^{-1}(b)}{(f ˆ g) Dg}. \end{aligned} \end{equation}

We can make a coordinate-independent notion of integration in the following way. An interval of the real line is a 1-dimensional manifold with boundary. We can assign a coordinate chart χ to this manifold. Let $x = χ(\mathsf{m})$. The coordinate basis is associated with a coordinate-basis vector field, here $\frac{∂}{∂ x}$. Let $\boldsymbol{ω}$ be a one-form on this manifold. The application of $\boldsymbol{ω}$ to $\frac{∂}{∂ x} is a real-valued function on the manifold. If we compose this with the inverse chart, we get a real-valued function of a real variable. We can then write the usual integral of this function

\begin{equation} I = ∫_a^b{\boldsymbol{ω}(∂ / ∂ x) ˆ χ^-1}. \end{equation}

It turns out that the value of this integral is independent of the coordinate chart used in its definition. Consider a different coordinate chart $x^′ = χ^′(\mathsf{m})$, with associated basis vector field $\frac{∂}{∂ x}$ . Let $g = χ^′ ˆ χ^-1$. We have

\begin{equation} \begin{aligned} ∫_a^{′}^{b^{′}} & \boldsymbol{ω}\left(∂ / ∂ \mathrm{x}^′\right) ˆ χ^′-1
&=∫_a^{′}^{b^{′}} \boldsymbol{ω}\left(∂ / ∂ \mathrm{x}\left(D\left(χ ˆ χ^′-1\right) ˆ χ^′\right)\right) ˆ χ^′-1 \ &=∫_a^{′}^{b^{′}}\left(\boldsymbol{ω}(∂ / ∂ \mathrm{x}) D\left(χ ˆ χ^′-1\right) ˆ χ^′\right) ˆ χ^′-1 \ &=∫_a^{′}^{b^{′}}\left(\boldsymbol{ω}(∂ / ∂ \mathrm{x}) ˆ χ^′-1\right) D\left(χ ˆ χ^′-1\right) \ &=∫_a^b\left(\left(\left(\boldsymbol{ω}(∂ / ∂ \mathrm{x}) ˆ χ^-1\right) D\left(χ ˆ χ^′-1\right)\right) ˆ g\right) D g \ &=∫_a^b \boldsymbol{ω}(∂ / ∂ \mathrm{x}) ˆ χ^-1, \end{aligned} \end{equation}

where we have used the rule for coordinate transformations of basis vectors (equation 3.19), linearity of forms in the first two lines, and the rule for change-of-variables under an integral in the last line.[fn:1]

Because the integral is independent of the coordinate chart, we can write simply

\begin{equation} I = ∫_\mathsf{M} \boldsymbol{ω}, \end{equation}

where $\mathsf{M} $is the 1-dimensional manifold with boundary corresponding to the interval.

We are exploiting the fact that coordinate basis vectors in different coordinate systems are related by a Jacobian (see equation 3.19), which cancels the Jacobian that appears in the change-of-variables formula for integration (see equation 5.2).

Higher Dimensions

We have seen that we can integrate one-forms on 1-dimensional manifolds. We need higher-rank forms that we can integrate on higher-dimensional manifolds in a coordinate-independent manner.

Consider the integral of a real-valued function, $\mathsf{f} : \mathsf{R}^n → \mathsf{R}$, over a region $\mathsf{U}$ in $\mathsf{R}^n$. Under a coordinate transformation $g : \mathsf{R}^n → \mathsf{R}^n$, we have[fn:2]

\begin{equation} ∫_\mathsf{U}{\mathsf{f}} = ∫_g^{-1(\mathsf{U})}{(\mathsf{f} ˆ g) det(Dg)}. \end{equation}

A rank $n$ form field takes $n$ vector field arguments and produces a real-valued manifold function: $\boldsymbol{ω}(\mathsf{v}, \mathsf{w}, \cdots, \mathsf{u})(\mathsf{m})$. By analogy with the 1-dimensional case, higher-rank forms are linear in each argument. Higher-rank forms must also be antisymmetric under interchange of any two arguments in order to make a coordinatefree definition of integration analogous to equation (5.3).

Consider an integral in the coordinate system $χ$:

\begin{equation} ∫_χ(\mathsf{U)}{\boldsymbol{ω}(\mathsf{X}_0, \mathsf{X}_1, \cdots) \ ˆ χ^-1}. \end{equation}

Under coordinate transformations $g = χ ˆ χ^′-1$, the integral becomes

\begin{equation} ∫_{χ^′(\mathsf{U})}{\boldsymbol{ω}(\mathsf{X}_0, \mathsf{X}_1, \cdots) \ ˆ χ^′-1 det (Dg)}. \end{equation}

Using the change-of-basis formula, equation (3.19):

Footnotes

[fn:2] The determinant is the unique function of the rows of its argument that i) is linear in each row, ii) changes sign under any interchange of rows, and iii) is one when applied to the identity multiplier.

[fn:1] Note $(D (χ ˆ χ^{′ -1}) ˆ (χ^′ ˆ χ^-1)) D(χ^′ ˆ χ^-1) = 1$. With $g = χ^′ ˆ χ^-1$ this is $(D(g⁻¹) ˆ g) (Dg) = 1$.