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.
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)?
@@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
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
Awesome content selection. Thanks!
Awesome overview!
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.
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
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!
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 😎
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
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 😎