(adsbygoogle = window.adsbygoogle || []).push({}); Integration in Sean Carroll's "parallel propagator" derivation

Reading Chapter 3 of Sean Carroll'sGeneral Relativity Lecture Notes, I've followed it up to and including eq. 3.38.

[tex]\frac{d}{d\lambda} P^\mu_{\;\;\; \rho}(\lambda,\lambda_0) = A^\mu_{\;\;\; \sigma} P^\sigma_{\;\;\; \rho}(\lambda,\lambda_0).[/tex]

Here, Carroll writes, "To solve this equation, first integrate both sides:

[tex]P^\mu_{\;\;\; \rho}(\lambda,\lambda_0) = \delta^\mu_\rho + \int_{\lambda_0}^\lambda A^\mu_{\;\;\; \sigma} (\eta) \; P^\sigma_{\;\;\; \rho}(\eta,\lambda_0) \; d \eta.[/tex]

"The Kronecker delta, it is easy to see, provides the correct normalization for [itex]\lambda = \lambda_0[/itex]."

I can see that this makes P the identity matrix in that case, but by what algebraic rule is it inserted. This is a definite integral, so shouldn't any constant of integration be cancelled out?

Also, I don't understand the iteration procedure that follows. Doesn't the concept of integration already encode such a procedure, taken to a limit? Should I read this as (coordinate-dependent?, coordinate independent?) abstract index notation for a matrix equation inside the integral sign, or is each component function integrated separately? Is there a name for this procedure or the subject area that includes it? Can anyone recommend a book or website that explains the mathematical background. Sorry these questions are a bit vague. I'm not really sure what to ask.

I wonder if it's related to what Bachman calls cells and chains. Maybe I should read the rest of that chapter first.

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# Integration in Sean Carroll's parallel propagator derivation

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