Parametric curves | Multivariable calculus | Khan Academy
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- เผยแพร่เมื่อ 25 ต.ค. 2024
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When a function has a one-dimensional input, but a multidimensional output, you can think of it as drawing a curve in space.
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Wait is this Grant Sanderson (3blue1brown)? That's awesome! LOL
Sounds like him
@@treasuresoffaith_zehrarisale It is him. He worked for Khan Academy for a while.
Wow. Now I hear that.
Sound like his sound
Grant, thanks for making these Multivariable Calculus videos. They're terrific.
really incredible skill with graphing but the volume keeps changing levels video to video
is this 3blue1brown?
Laurent Garcia yup
I know I was thinking the same thing
lmao
Yesassssss
no, this is Patrick
the concept of input space and output space seems to be key here. we are so used to seeing input and output space on the same chart that separating the two may feel alien at first.
Khan academy, will you soon start teaching abstract/pure mathematics? Its hard to find easy resources online for these subjects
extremely lol
nice
Books
That would be awesome
If grant was my professor I’d never skip lecture!
Grant, thank you. What I am even more interested in is how do you draw these cool graphics.
python
Grapher, not phyton
Desmos calculator
U should check out his own channel, 3blue1brown there is even better animations
I was using t= 2(n)/pi that generate exact same graph instead of using 0
ITS GRANT!!!
Awesome method of teaching. Terrific work.
You made the world a better place to live in...this might lead to find a new world to live in....
At 5:43, shouldn't t=1 still be in the first quadrant? After the origin, the first time it crosses the y=axis is at t=pi/2,isn't it?
Yes. You are absolutely correct. If you plug in the value t = 1 at (tcost, tsint) you get something like value (0.54, 0.84) which indeed is the first quadrant. And yes the first time the spiral hits the y axis is at t=pi/2 and then at t = pi + pi/2 and then at
t = 2*pi + pi/2 and so on.
Thanks, I'm not crazy. Such a great explanation, shame to have a small slip up there (would be nice if he just put a note over the video to update it)
No, he said that he is describing a different function that gives the same curve as (tcost, tsint).
Such a soothing voice
3Blue1Brown is his channel name.
hi @@CJBurkey are you alive
شكرا
thank you so much
No way he is grant!!
Great video
Cool spiral! 😊
Calling them "pi-halves" is so cute!
It's easier than what it sounds
thank you so much!
Marvellous💯
the best source
Which tool are you using to draw these parametric functions?
What is the function that draws the same graph??
The plot of t*cos(t), t*sin(t) is equivalent as if you drew a circle with a compass whose legs are being constantly stretched.
There's a type of parametric curve everyone has missed. graph a slice of a circle no bigger than a semicircle then change one of the radius lengths. Now progress the one radius length to the other radius length in the simplest linear way you should find that for any triangle every side has a corresponding curve. In 3D this also works for curved surfaces on the faces of any tetrahedron. Not only that but you can change how the one radius progresses to the other radius and this even works in 3D if you want to say add bumpiness or roughness to a human face or want to flatten the curvature a bit to represent well the face of wall. Not only this but the surface setting on the triangle can also represent coloring of the surface. In terms of video compression this technique can allow for a very upscalable image be the curvature of smaller detail not as advanced you could also save a lot on data size of the video stream.
What is the second function?
Can this explain the shape of galaxies?
what software do you use to plot these? Excel or something else?
Hey! That's 3Blue!Brown!! 😁
parameter is just kind of a fancy word for input
How pi/2 is 1.7?
Probably didn't convert pi into ~3.14, probably thought about the common sinusoidal output 1.73~sqrt(3)/2 which seems "about right." I.e. the Mandela effect. Even professors at top universities make silly mistakes, particularly arithmetic ones, probably because they fear embarrassment.
pls someone should help me out with this assignment
Calculate the frenet apparatus for the parameterized curves alpha(t) = (3t-t³,3t²,3t+t³)
wrg, no problx about that, can do anyx
Can we say that F(t)=F[x(t),y(t)] in a two dimensional cartesian system?
I don't see why not we would only have one parameter that is T and we can use it to graph the x axis as f(T) and y axis as g(T) and then we can represent it as T changes with linear speed from say 0 to your particular value. Sure it may be a bit hard to visualise where what value of T turns us this particular value but that's the tradeoff we have to do. Even in this video you will realise that we do not know what value of T does to the value of T into sin of T as x and T I to cos of T as Y axis. Just like our previous example we do not know how much T is required for the vector to reach particular value but we can visualise it.
is it just a coincidence that this looked like the golden spiral?
dear sir on which axis 't' value has to take?
t is not shown because hes only showing the output of the function, have a look at the videos before this in the playlist
How π/2 is 1.7?????
Vocie of 3blue1brown
hi
3 brown 1 blue voice?
Holy crap you sound so different! I'm on my phone so I can't see the date...but if I had to guess, this video is >10 years old
Alex Merical No, this is grant from 3b1b, not sal
I dont have Ocd, but come on man! Take an extra second to draw something more like a circle! What even is that! D,: that's on the same level as taking a bite of a KitKat without breaking them apart first, it's so Antisatisfying to look at
Is there a reason you call it "parametric function" instead of path/curve? From my experience thats not only the denomination always used in maths but also sounds a lot more intuitive and less intimidating. Maybe its just this way in my classes though but the only time I ever heard "parametric function" was in my programming lessons.
Edit: Ok now I feel stupid. THe domain of curves and paths are just intervalls aren'T they? I guess parametric functions can have a domain in the whole real numbers?
A parametric function of 1 parameter is a curve, and for 2 parameters it's a surface.
In math definition, parameter is defined as a set of selected values with some limits to them (in other word they have interval, or maybe they're chosen one by one without specifying interval), so parametric function is a function that relies on those selected values/interval (the parameter)
Why is cos(0) = 0 in this? my life makes no sense .....
It's not "cos(0) = 0", it's "0 * cos(0) = 0". Zero multiplied by anything is zero, nothing to be surprised at.