From the point of view of a person who does somewhat understand how this kind of thing works in algebraic topology, this was interesting to see how it can be done from the calculus viewpoint. It is pretty much the foundation of algebraic topology, though, and as such that field is a lot more suited for this. A bit more complicated, though, and I may be a little biased when talking about the stuff I like. Anyway, you are doing superb work in popularising math, so keep it up. :)
We just covered winding numbers in my honors complex variables class, and its cool to see it from a multivariable side of things. Although, instead of winding number around 0, its around any z0. Ie instead of 1/z its 1/(z-z0). Have a feeling the real case will come in handy during my differential geometry class next semester, could be wring tho lol.
dr peyam, does it make sense to speak of a winding number of a curve around a point in 3 dimensions or is it soley a 2 dimensional concept? (say we are given a point P=(0,0,1) and four curves: (cos(t), sin(t), 0), (cos(t), sin(t), (1-cos(t))/2), (cos(t), sin(t), 1-cos(t)) and (cos(t), sin(t), 1), where 0
At around 4:00, it is mentioned that the integral over the gradient of the arctan function will yield the angle at the end minus the angle at the beginning. For a non-conservative vector field, does this still hold?
@@PS-dw5qo We're actually exploiting fact that this vector field is non-conservative. If it was conservative the integral would vanish which is not what we want.
Hello, Dr Peyam, I have been trying to find a solution for computing the number of rotations a cable of L (say 80 m) and thickness of 4cm, when wound around a circle of 80 cm. I have been trying to invent formulas to show how many rotations you will need to fully wind the cable around the 80 cm circle - it turns out that I am not Gottfried Leibnitz sadly! Would you have any idea of the formula needed for an accurate result? Many thanks in advance if you are reading this! (I know Its connected to this formula but the dimension of thickness needs some differential equation I cant guess at!)
The imaginary term does not dissapear for general planar curves, it is equal to integral[ ln( r ) dz] :) Also we can find the result quickly since integral[ (1/z) dz] = ln(z) = ln(r) + i(theta). And so with bounds on the integral the imaginary component is the difference in the ending and beginning angle values theta.
Hello, what is the winding number of a vector field? We only talked about the winding number of a curve, and we only talked about vector fields to integrate over them.
@@Craynz I wrote a polyline class (with derived classes polyarc and polyspiral) with a method "in" which tells whether a point is in the curve. To handle spirals, I had to make it recursive, and therefore additive. This meant that I had to use the winding number and define the value of the winding number at a point on the curve. I decided that the winding number at a point on a curve is the limit as r->0 of the average winding number at all points inside a circle of radius r centered at the point. This makes the winding number at a point on a circle 1/2, but at a point on the snowflake the limit does not exist. As to the center of the snowflake, the winding number is 1, but cannot be computed with calculus, as the function that traces out the snowflake is not differentiable.
CrayNz this is a bit of dangerous thinking, because just because a point on the curve isn’t an endpoint of one of the segments, I’m not sure that means that the curve is differentiable at that point, because the curve itself is the limit of piecewise linear curves, but differentiability is not preserved under limits.
pr peyam, could you make a series about calculating probabilities, expactations and variances of coin tosses, dice rolls, card deals, choosing balls of various colours out of the bowl, and two or max three times branching processes involving the previous ones?
What is your definition of a conservative vector field? Mine is that the integral around any loop is trivial. And so /(x^2+y^2) would not be. This doesn’t contradict the fact that gradients are always conservative, because the function you’re taking the gradient of is theta, which is not actually a well-defined function. On the other hand, the other piece of the complex winding number you had is the gradient of ln(x^2+y^2), which is well-defined. So that’s why its integral is always 0.
What makes my day... A video starring with : " Thanks for watching and today..... "
Thank you for the video. Now I understand that hard part from lecture, where that wasnt explained as such intuitivly
Thanks for your clear explanation.
From the point of view of a person who does somewhat understand how this kind of thing works in algebraic topology, this was interesting to see how it can be done from the calculus viewpoint. It is pretty much the foundation of algebraic topology, though, and as such that field is a lot more suited for this. A bit more complicated, though, and I may be a little biased when talking about the stuff I like. Anyway, you are doing superb work in popularising math, so keep it up. :)
Thank you :)
A rainy day here in Argentina with math, the best thing!! Thanks for always bringing awesome content 😄
We just covered winding numbers in my honors complex variables class, and its cool to see it from a multivariable side of things. Although, instead of winding number around 0, its around any z0. Ie instead of 1/z its 1/(z-z0). Have a feeling the real case will come in handy during my differential geometry class next semester, could be wring tho lol.
dr peyam, does it make sense to speak of a winding number of a curve around a point in 3 dimensions or is it soley a 2 dimensional concept? (say we are given a point P=(0,0,1) and four curves: (cos(t), sin(t), 0), (cos(t), sin(t), (1-cos(t))/2), (cos(t), sin(t), 1-cos(t)) and (cos(t), sin(t), 1), where 0
I haven’t heard about 3D generalizations, but I’m sure there must be some! One idea is to have 3 winding numbers, one in the xy plane, etc
At around 4:00, it is mentioned that the integral over the gradient of the arctan function will yield the angle at the end minus the angle at the beginning. For a non-conservative vector field, does this still hold?
Over a circle around the origin the vector field will not be conservative. How should one interpret the integral in that case?
@@PS-dw5qo We're actually exploiting fact that this vector field is non-conservative. If it was conservative the integral would vanish which is not what we want.
Excellent ! For me a relation between the length of a function and windings numbers. Thank you very much !
Amazing ! Thanks a lot. Could you provide a proof for the general case you mention ? I mean for curves without loops
Good to know! Thank you, Peyam.
Can a winding number be related to the periodicity of the curve (in polar coordinates)?
Hello, Dr Peyam, I have been trying to find a solution for computing the number of rotations a cable of L (say 80 m) and thickness of 4cm, when wound around a circle of 80 cm. I have been trying to invent formulas to show how many rotations you will need to fully wind the cable around the 80 cm circle - it turns out that I am not Gottfried Leibnitz sadly! Would you have any idea of the formula needed for an accurate result? Many thanks in advance if you are reading this! (I know Its connected to this formula but the dimension of thickness needs some differential equation I cant guess at!)
The imaginary term does not dissapear for general planar curves, it is equal to integral[ ln( r ) dz] :)
Also we can find the result quickly since integral[ (1/z) dz] = ln(z) = ln(r) + i(theta). And so with bounds on the integral the imaginary component is the difference in the ending and beginning angle values theta.
dr peyam, is there a relation between a winding number of a vector field and its curl?
Probably! Via Stokes’ theorem maybe :)
Hello, what is the winding number of a vector field? We only talked about the winding number of a curve, and we only talked about vector fields to integrate over them.
(8:56)
society: aaghh!! what a freak!! X(
mathematicians: uuu!! what a beautiful freak!!
Wow! That thing is the logarithmic residue for f(z)=z!I wasn’t told that in uni, thanks:)
What is the winding number of
*a circle around a point on the perimeter?
*a snowflake around the center?
*a snowflake around a point on the perimeter?
@@Craynz I wrote a polyline class (with derived classes polyarc and polyspiral) with a method "in" which tells whether a point is in the curve. To handle spirals, I had to make it recursive, and therefore additive. This meant that I had to use the winding number and define the value of the winding number at a point on the curve. I decided that the winding number at a point on a curve is the limit as r->0 of the average winding number at all points inside a circle of radius r centered at the point. This makes the winding number at a point on a circle 1/2, but at a point on the snowflake the limit does not exist.
As to the center of the snowflake, the winding number is 1, but cannot be computed with calculus, as the function that traces out the snowflake is not differentiable.
CrayNz this is a bit of dangerous thinking, because just because a point on the curve isn’t an endpoint of one of the segments, I’m not sure that means that the curve is differentiable at that point, because the curve itself is the limit of piecewise linear curves, but differentiability is not preserved under limits.
pr peyam, could you make a series about calculating probabilities, expactations and variances of coin tosses, dice rolls, card deals, choosing balls of various colours out of the bowl, and two or max three times branching processes involving the previous ones?
Not sure, but that’s definitely something I might consider! I think blackpenredpen has videos on this
@@drpeyam thank you, I will check him out.
That’s the one I found, at least: m.th-cam.com/video/HUtHZyBsUnA/w-d-xo.html
What is your definition of a conservative vector field? Mine is that the integral around any loop is trivial. And so /(x^2+y^2) would not be. This doesn’t contradict the fact that gradients are always conservative, because the function you’re taking the gradient of is theta, which is not actually a well-defined function. On the other hand, the other piece of the complex winding number you had is the gradient of ln(x^2+y^2), which is well-defined. So that’s why its integral is always 0.
My definition is that F is the gradient of a function
Dr Peyam Ok, then again this vector field is not conservative because theta is not a function.
I watched t-shirt tag all the time