Deriving the Arc Length in Cartesian and Polar Coordinates
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- Опубліковано 12 вер 2024
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Let us go ahead and get started! Starting from the riemann sum definition of a line integral, we can actually evaluate the arc length of ( polar ) curves. Feel free advancing this idea to different coordinate systems, such as cylindrical, spherical and toroidal ones!
ENjoy! =)
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![Deriving the Arc Length and Area of a Cycloid! [ Cut Version ]](http://i.ytimg.com/vi/dTBMRnhgbhI/mqdefault.jpg)
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Some people were straight before they found your channel
haha XD
so you saying he is so ugly that he turns girls into lesbians ? dude, that's rude
@@Cashman9111 no turns dudes to dudees
@@Cashman9111 maybe that means he is so handsome that he turns men into gays
Now I want to do this for curved spacetime.
If you define your manifold with a metric then it's practically in the definition. If you start from a connection and no metric, then you can't (unless you assume metric compatibility)
This is what a physicist really likes to do😀
Get a load of this guy!
I zoomed in very very close and now I'm straight! Thanks papa!
Once again Leibnitz notation proves its superiority.
Well, if you zoom in everything is straight😂😂😂
I'M NOT STRAIGHT
* sees video *
OK MAYBE I'M STRAIGHT
Oh boi I love this homophobic humor
please give me your definition of homophobia and not the one made up by second rate humies (i.e. SJWs)
@@mononix5224 Aversion to and discriminations against homosexuality/homosexual people? What is the "made up" one? Because all definitions are made up by somebody...
*Homomorphic
what are the odds that literally today I was assigned to derive arc length in polar coordinates for hw, I love you
I don't know you, but we have the same taste in music.
lol i watched this last week, here i am again, because deriving arclength in polar coords is a homework problem. big happy :'D
nice XD
Great timing papa flammy, I was about to try and manually reverse my sexuality without any help
May i just say... you are legendary! Love all your meme sentences hahaha
:))
Okay, this is epic❤️👌
More Physics 😂😂
10:15 good that it is the same Spiel ^^
great to see a young man just being himself !!
:)
First. Definitely first.
And this finally seems like within my area of current knowledge!
You explain what your doing really well.
Great video! Do you have a video formalizing the concept of "playing physicist" i.e. some sort of rigorous proof that we can "always" formulate proofs using dy's, dx's etc. without the need to mess around with Riemann sums and limits?
Sometimes I don't know what you're talking about, but I still watch it to feel smart lol
1:16 Awaapabaund 😍🥰
Im researching the derivation of arc length for polar equations, when i found out this video. My laught on that moment. Hahahaha. Nice one 😂 i really enjoyed your video so much. Lovelots from the Philippines 😆
Papa Flammy has mastered the sacred ancient art of the Chen Lu
Guten Tag meister Sholze ; *please show us how is the Lagrangian formulation , if the fixed limits are not points in two dimensions ; but are lines in the three dimensional space*
*In this case , what would be the """ minimum sheet """???*
I would be very gratefull .
Greetings from Brazil !!!
Ray Viana Sampaio .
chen lu rule !¡!¡! yeaah papa!
When i grow up i dont want to get the rank of sir or doctor but i want to get the rank of PAPA
This is perfect timing papa, we just learned about arc length in class today. Thank you my boi
What is delta L. Shouldn't it be just L. The length is the space cut from a point to another. So, what you call delta L is the length and not the change in it.
Charlie: :P
Papa: "I'm looking at you my boy"
Charlie: :o
The face when you're gay and named charlie watching papa flammy D:
@@PapaFlammy69 lol I have a rare-ish name, it was just really surprising. the subset of people must be very small: gay, named charlie, interested in math and science.
Who ever said germans don't have humor??? AJAHAHA that fucking killed me
As you said in 1:58 that we are going to do an approximation I felt like an engineer
OMG PAPA FLAMMY! IT'S WAS AMAZING!
Papa! When substituting the Cartesian coords with the polar ones shouldn't they be x ( t ) = r(t) * cos ( theta( t ) ) and likewise y ( t ) = r (t ) * sin ( theta ( t ) ) ? Luv ur videos, keep up the good work!
Papa flammy is kazuma!!!
why the arc length formula( cartesian form) cannot be used to find the arc length
Have you used Riemannian manifold to define this things?
A true trip. Thank you so much
Question: Shouldn't you say theta(r) in the argument of the trig functions when converting from cartesian to polar? The reason why I say this is because in polar sometimes the angle changes as well.
9:15 I believe you confused theta with t. Shouldn't it be x(t) = r(t) cos(theta(t)) usw.?
Besides I think I'm still straight I love you Papa! You're one of [my] heroes!.....
Wouldn't it be fun to make a video of deriving the formula of the arc area in Cartesian, spherical and cylindrical coordinates?
Not for clarification purposes, it was for obfuscation purposes.
Could you show us the contour special case from the beginning nextly?
Cool, but can you solve the integral from 0 to infinity of 1/(x^n + 1)? I started trying it.
Dunno if this helps but I have a contour that’s a sector with angle 2*pi*n/2 with radius R as it approaches infinity. It’s not as messy as I thought.
9:43 x dot...
🙏
Min 9:49 - Polar Coordinates: It´s not the chain rule but the product rule.
What is triangle area when its equal to parameters of area cover by function in any interval
??????????
We need to know how hypotenuse of triangle regulated its position or unit circle multiple times the area of function under any interval ?
How unit circle area equal to function under curve in a given interval
So we found a simple percentage ???
Papa I have a question, where does the integration limits move for the polar coordinates?
Oh nice! I learned about this the last week :D only 5 years left to start grasping your math knowledge
But after 5 years you will have a 5years more knowledge than me... Oh welp at least I'm not going backwards lol
Btw what is the rigorous relation between dl and dx,dy? I'm more interested in the rigorous way to deal with differential than actually coming up with normal formulas
Make a video deriving the Covariant Derivative (Riemann Geometry)
That's... A bit tougher than anything this channel usually covers. A fairly easy extrinsic definition of the Covariant derivative though is to embed your manifold in R^n (carrying the tangent bundle via the embedding in the natural way) and then orthogonally project down the tangent bundle in the sense that each tangent space in R^n gets orthogonally projected down to a tangent space on the embedded manifold.
Intrinsic definitions can be nicer to actually work with, and the most common one involves two terms: Dv/dt = dv^k/dt d/dx^k + v^i dp^j/dt Christoffel(i, j, k) d/dx^k. In the intrinsic definition, the first term is what you would get if we were working over R^n or any "flat" space, while the second term is a correction that takes into account how our space is "curved". This is important since when we define the derivative we want it to act like an infinitesimal difference quotient, but nearby tangent spaces might "twist away" from each other if our manifold is curved (think of taking a block of rubber in both hands and twisting each hand in opposite directions. The rubber becomes twisted. The technical name for this is "Torsion" and is a tensor).
Thanks papa
Shouldn't x(t) and y(t) functions of r(t) and theta(t)? Like r(t)*cos(theta(t))?? @flammie
I derived this result on my own when I was in 9th grade after my friend challenged me ;)
So r is the vector to any point on the curve?
How could dt go with the absolute value
(Units don't match in final formula). Should be using x(t)=r(t) cos(theta(t)) -> xdot(t)=rdot(t) cos(theta(t))-r(t) sin(theta(t)) thetadot(t)
You can think of t being proportional to θ, or that it's sweeping with constant angular speed.
@@badrunna-im need angular frequency in from of the term. theta = omega * t to make it unitless ;)
I wrote same Spiel on my exam sheet. Now I am going to invade engineering faculty with a professor.
Are you trying to become a professor
You are funny. Good explanation. Hope UA-cam don't censor your video. A fresh breath among the politically correct speeches.
Hi flammy ❤❤❤
Nice 👍🙂
Wait, you cover this in calc 1?
Yea lol this was Calc three stuff for me
Shouldn’t it be theta(t)
Wouldn't it be sqrt((r*dtheta/dt)^2 + (dr/dt)^2)?
And... x = r(t)*cos(theta(t))
y = r(t)*sin(theta(t))
No, t *is* theta in this case
Doing this with vectors in calc 3 rn. Not a fan of the new material lol
Soo... a circle is the gayest shape.
Am I the only one wanting him to derive something in toroidal coordinates?
i think u r teaching in a higher level
Oh yeah yeah
I'm gay and I find you really cute.
9:47 *product rule :p
I only know the Chen lu no chain rule
Hi derive man
lets play physicist lmao
If physicists approximate, why do they make fun of engineers because of this?
Because stereotypically physicists approximate in a more rigorous way than engineers.
Because we are correct 😉 A&Ω
Because engineers shoot for +-30% so they can charge again later for the "redesign under new customer parameters" when it doesn't work. That's why engineering degrees require an Ethics class and degrees in Physics do not.
Engineers approximate physicist's approximations because they test the practicality of the theory and throw out the rest. But then again looking at field theory and relativity you get some heavy approximations that drive all research centers even today
because Pi=e=3
vruh
Chen lu😂
Lies!
hahaha how is homosexuality even real hahaha just zoom in lol
Homosexuality is not something that needs to be "cured"!
It's obviously a joke, I'm fine with the video, though the comments are confusing in parts because some come across as actually homophobic and still got a like from him - maybe it's all ironic and joking, but I feel like if the joke becomes indistinguishable from actual homophobia it might not be a good joke.
You are skinny.
First
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