Thank you for an amazing and in depth explanation of trigo fundamentals. I was revisiting calculus after realising I learnt nothing in school. This part was really tough to understand by just reading a book. Please keep updating such wonderful content.
Thinking about this, using archimedes first axiom, we can prove the current limit definition will be a lower bound to the curve's actual perimeter (because we're always connecting points on the curve on the limit, their length is necessarily shorter). I dont know if we can prove using archimedes second axiom that the limit definition is an upper bound as well, but it'd lower my philosophical worries (although archimedes 2nd axiom is in no way as clear as the first one)
Oh no! I was sooo enjoying this series until I realized that I have no clear definition of a straight line. :-( (I was totally happy with the definition of a straight line being shortest. :-( :-( )
He was the first to compute some curved lengths, areas, and volumes, but note that he never worked with a metric or line element even in the flat case. The comparison to the style of Riemann's analytical work is fair, though, and so impressive because Archimedes was working more than 2000 years earlier.
Thank you for an amazing and in depth explanation of trigo fundamentals. I was revisiting calculus after realising I learnt nothing in school. This part was really tough to understand by just reading a book. Please keep updating such wonderful content.
Glad it was helpful!
Thinking about this, using archimedes first axiom, we can prove the current limit definition will be a lower bound to the curve's actual perimeter (because we're always connecting points on the curve on the limit, their length is necessarily shorter). I dont know if we can prove using archimedes second axiom that the limit definition is an upper bound as well, but it'd lower my philosophical worries (although archimedes 2nd axiom is in no way as clear as the first one)
This is an excellent video. I especially appreciate how you point that using areas of triangles to prove cos x
Great material! I like how you motivate the steps. I learned a lot.
what is the deepest extent we can go to in order to prove x>sinx .??????
Oh no! I was sooo enjoying this series until I realized that I have no clear definition of a straight line. :-( (I was totally happy with the definition of a straight line being shortest. :-( :-( )
Looks like Archimedes was the first Riemannian geometer.
He was the first to compute some curved lengths, areas, and volumes, but note that he never worked with a metric or line element even in the flat case. The comparison to the style of Riemann's analytical work is fair, though, and so impressive because Archimedes was working more than 2000 years earlier.
@@DanielRubin1 Yea I was moreso thinking about the flavour of his work. Perhaps it's more accurate to say he was the first analyst.
Well videos are too fast