# The Unapologetic Mathematician

## Integrals are Independent of Parameterization

If $c:[0,1]^k\to M$ is a singular $k$-cube and $\omega$ is a $k$-form on the image of $c$, then we know how to define the integral of $\omega$ over $c$:

$\displaystyle\int\limits_{c([0,1]^k)}\omega=\int\limits_{[0,1]^k}c^*\omega$

On its face, this formula depends on the function $c$ used to parameterize the region of integration. But does it really? What if we have a different function $d:[0,1]^k\to M$ with the same image? For convenience we’ll only consider singular $k$-cubes that are diffeomorphisms onto their images — any singular $k$-cube can be broken into pieces for which this is true, and we’ll soon deal with how to put these together.

Anyway, if $c([0,1]^k)=d([0,1]^k)$, then given our assumptions there is some diffeomorphism $f=c^{-1}\circ d:[0,1]^k\to[0,1]^k$ such that $d=c\circ f$. If $J_f$ is everywhere positive, then we say that $d$ is an “orientation-preserving reparameterization” of $c$. And I say that the integrals of $\omega$ over $c$ and $d$ are the same. Indeed, we calculate:

$\displaystyle\begin{aligned}\int\limits_{c\circ f}\omega&=\int\limits_{[0,1]^k}(c\circ f)^*\omega\\&=\int\limits_{[0,1]^k}f^*c^*\omega\\&=\int\limits_fc^*\omega\\&=\int\limits_{f([0,1]^k)}\left[c^*\omega\right]\left(\frac{\partial}{\partial u^1},\dots,\frac{\partial}{\partial u^k}\right)\,d(u^1,\dots,u^k)\\&=\int\limits_{[0,1]^k}\left[c^*\omega\right]\left(\frac{\partial}{\partial u^1},\dots,\frac{\partial}{\partial u^k}\right)\,d(u^1,\dots,u^k)\\&=\int\limits_c\omega\end{aligned}$

where we use our expression for the integral of $\omega$ over the image $f([0,1]^k)$ in passing from the third to the fourth line. Thus the integral is a geometric quantity, depending only on the image $c([0,1]^k)$ and the $k$-form $\omega$ rather than on any detail of the parameterization itself.

About these ads

### Like this:

Like Loading...

Posted by John Armstrong | Differential Topology, Topology

## 2 Comments	»

1. [...] it all works out for the same reason parameterization invariance and the change of variables formula do. Passing from the boundary of the singular cube back to the [...]

Pingback by | August 20, 2011 | Reply

2. [...] we use to reparameterize our integral. Of course, this function may not be defined on all of , but it’s defined on , [...]

Pingback by | September 5, 2011 | Reply

« Previous | Next »

## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

• ## RSS Feeds

RSS - Posts

RSS - Comments

• ## Feedback

Got something to say? Anonymous questions, comments, and suggestions at Formspring.me!
%d bloggers like this:
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 27, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8871572613716125, "perplexity_flag": "head"}