I find this concept, and the related Radon Nikodym derivatibe unintuitive - this helps. One area where the motivation works better is in probabily theory where the ingegral of g(x) dF(x) where F is cumulative proabality function is the integral g(x) f(x) where f(x) is the probability density function - namely the expectation of g(x)
If the function g is a parameterization of a curve, then wouldn't the Riemann-Stieltjes integral become a line integral? Would it be incorrect to interpret the line integral as a special case of the Riemann-Stieltjes integral?
@@randomdude8171 Yes, if g is a parameterization of a curve, the RS integral becomes a line integral. In fact, line integrals can be considered as special cases of the RS integral, where g describes the path and f represents a scalar field evaluated along the curve. So, it’s not incorrect to interpret the line integral this way. The RS integral is very flexible, and different interpretations depend on how g is chosen.
@@apolloandartemis4605 The intuition behind the RS integral relates to how the increments of the function g weight the values of f as you sum over intervals. This can be connected to u-substitution, because the change in variables modifies the integration scale, which is how the differential dg changes in the RS sum, which “re-weights” contributions at different points. Both involve adjusting the way we sum contributions to account for a variable transformation
Thanks for this! I encountered some integrals with respect to a function in probability but was not familiar with these. So is it correct to say that if f(x) = x, they the integral of x with respect to g(x) is the area under g-inverse(x)?
I find this concept, and the related Radon Nikodym derivatibe unintuitive - this helps. One area where the motivation works better is in probabily theory where the ingegral of g(x) dF(x) where F is cumulative proabality function is the integral g(x) f(x) where f(x) is the probability density function - namely the expectation of g(x)
I'm starting to think, Riemann is a criminally underrated mathematician by the general public.
@@sphakamisozondi yes, you’re right
If the function g is a parameterization of a curve, then wouldn't the Riemann-Stieltjes integral become a line integral? Would it be incorrect to interpret the line integral as a special case of the Riemann-Stieltjes integral?
@@randomdude8171 Yes, if g is a parameterization of a curve, the RS integral becomes a line integral. In fact, line integrals can be considered as special cases of the RS integral, where g describes the path and f represents a scalar field evaluated along the curve. So, it’s not incorrect to interpret the line integral this way. The RS integral is very flexible, and different interpretations depend on how g is chosen.
Hi! How does this relate to the geometric intuition behind u-sub?
@@apolloandartemis4605 The intuition behind the RS integral relates to how the increments of the function g weight the values of f as you sum over intervals. This can be connected to u-substitution, because the change in variables modifies the integration scale, which is how the differential dg changes in the RS sum, which “re-weights” contributions at different points. Both involve adjusting the way we sum contributions to account for a variable transformation
@@dibeosthank you so much! Love the videos by the way.
@@apolloandartemis4605 thanksssss 😎
Thanks for this! I encountered some integrals with respect to a function in probability but was not familiar with these. So is it correct to say that if f(x) = x, they the integral of x with respect to g(x) is the area under g-inverse(x)?
@@pandabers no, the area is the projection of g(x) under f(x) on the f-g plane
Awesome overview!
Awesome content selection. Thanks!
wake up babe the integral vid is out.
@@rewixx69420 😎😎😎😎😎
@@dibeosseria bueno, si lo podrían traducir a español 👍🏻
@@renengan25 ¡Nuestro objetivo futuro es hacer todos estos videos en español, ya que es el segundo idioma más hablado en el mundo!
pretty nice moring
i hope this wont get me vomting....... although keep these things up
@@JOHN-ex8rb you can do it. I believe in you 😎
Interesting video!)
@@SobTim-eu3xu we are glad that you liked it!
@@dibeos like I amazed by animations, and animation of drawing line, like its not straight but wobbly, I love that one