I How do a bunch of integrals make an n-simplex or an n-cube?

[George Keeling]

TL;DR

Problem with series of integrals in parallel propagator. Parallel transporting vectors.

This question arises from Carroll's Appendix I on the parallel propagator where he shows that, in matrix notation, it is given by$$P\left(\lambda,\lambda_0\right)=I+\sum_{n=1}^{n=\infty}T_n$$and$$T_n=\int_{\lambda_0}^{\lambda}{\int_{\lambda_0}^{\eta_n}\int_{\lambda_0}^{\eta_{n-1}}{\ldots\int_{\lambda_0}^{\eta_2}A\left(\eta_n\right)A\left(\eta_{n-1}\right)\ldots A\left(\eta_1\right)}d^n\eta}$$$$\frac{1}{n!}\int_{\lambda_0}^{\lambda}{\int_{\lambda_0}^{\lambda}\int_{\lambda_0}^{\lambda}{\ldots\int_{\lambda_0}^{\lambda}\mathcal{P}\left[A\left(\eta_n\right)A\left(\eta_{n-1}\right)\ldots A\left(\eta_1\right)\right]}d^n\eta}$$where $\mathcal{P}$ orders the matrices $A\left(\eta_i\right)$ so that $\eta_n\geq\eta_{n-1}\geq\ldots\geq\eta_1$.

Carroll says that the $n$th term in the first series of integrals is an integral over an $n$-dimensional right triangle or $n$-simplex:$$T_1=\int_{\lambda_0}^{\lambda}{A\left(\eta_1\right)d\eta_1}$$$$T_2=\int_{\lambda_0}^{\lambda}{\int_{\lambda_0}^{\eta_2}A\left(\eta_2\right)A\left(\eta_1\right)d\eta_1d\eta_2}$$$$T_2=\int_{\lambda_0}^{\lambda}{\int_{\lambda_0}^{\eta_3}\int_{\lambda_0}^{\eta_2}A\left(\eta_3\right)A\left(\eta_2\right)A\left(\eta_1\right)d^3\eta}$$and illustrates them thus:

I can see how a 2-simplex with short sides equal is half a square and how that carries on for more dimensions but I can't see how the expression, for example, for $T_2$ is an integral over a right angle triangle, let alone one with short sides equal $\lambda-\lambda_0$.

It would seem that what exactly $A$ is is immaterial. I tested calculating $T_2$ both ways for three cases$$A\left(x\right)=3x^2,A_{\ \ \rho}^\mu\left(\lambda\right)=\left(\begin{matrix}0&\sin{\psi}\cos{\psi}\\-\cot{\psi}&0\\\end{matrix}\right),\ A_{\ \ \rho}^\mu\left(x\right)=\left(\begin{matrix}2x&3x^2\\4x^3&5x^4\\\end{matrix}\right)$$the first two gave the same answer by both methods and the third did not (I may easily have made a mistake on that.) The second is what $A_{\ \ \rho}^\mu$ would be for propagating a vector on the surface of a sphere at constant latitude $\psi$. It was much easier to use the second method (over an $n$-cube) and that did indeed parallel propagate vectors round lines of constant latitude very well.

If anybody can help me understand why the integrals are integrals over $n$-simplices or $n$-cubes and how $\mathcal{P}$ orders the matrices $A\left(\eta_i\right)$ so that $\eta_n\geq\eta_{n-1}\geq\ldots\geq\eta_1$ (the $\eta_i$ seem to be moving targets) I would greatly appreciate it.

 

[ergospherical]

I can't see how the expression, for example, for $T_2$ is an integral over a right angle triangle, let alone one with short sides equal $\lambda-\lambda_0$.

The long side of the triangle has the equation $\eta_1 = \eta_2$, so to calculate the multiple integral over the triangle you first hold $\eta_2$ constant and then integrate over $\eta_1$ from $\lambda_0$ to $\eta_2$ (from the left edge to the long side), and then you integrate the result over $\eta_2$ from $\lambda_0$ to $\lambda$.

 

[George Keeling]

Thank you @ergospherical I think I understand now. The multiple integrals over the square are more readily comprehensible.