loud goblet
I have a few questions on the things we learned, because I don’t really know where they come from:
They just gave us this and we just have to accept it but I didn’t understand from where it comes from.
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I have a few questions on the things we learned, because I don’t really know whe...
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surreal ember
The second condition doesn't seem correct. Shouldn't it be F(2a - x, 2b - y) = F(x, y), rather?
Not sure about the third, though.
loud goblet
The second condition doesn't seem correct. Shouldn't it be F(2a - x, 2b - y) = F...
It’s what they wrote so I am not sure about it either. I was also trying it at an exercise and I kept getting something wrong
surreal ember
Well, the 2s should definitely be there, as then this becomes an identity for (x, y) = (a, b). Otherwise it doesn't quite work out.
loud goblet
Well, the 2s should definitely be there, as then this becomes an identity for (x...
yes ur right
I just trried it
and I got that
but from where does this property come
surreal ember
Oh, basically, you know how if you take a vector and change the signs of all of its coordinates, then it just inverts about the origin?
loud goblet
yes
surreal ember
This is basically the same property, just with some translation (since the center may not be at the origin).
loud goblet
(x,y,z) -> (-x,-y,-z)
pine cypress
Imagine (a,b) is the center of symmetry, then if you take the function F(x-a,y-b), it will be symmetric around the origin, right?
Cuz it's only translation
And then since it's symmetric, F(x-a,y-b)=F(a-x,b-y)
And then you can de-translate and get
F(a-x+a,b-y+B)
Small b thefe
And there you see where the 2 comes from
surreal ember
Yeah, exactly.
pine cypress
(x,y)→(x-a,b-y)+(a,b)→(a-x,b-y)+(a,b)→(2a-x,2b-y)
surreal ember
Oh, actually, I do know how we can get the third property, assuming F is smooth enough to have those first derivatives.
So, we have F(x, y) = F(2a - x, 2b - y), right? Try differentiating both sides with respect to x.
loud goblet
So, we have F(x, y) = F(2a - x, 2b - y), right? Try differentiating both sides w...
okay give me a sec
loud goblet
Oh, actually, I do know how we can get the third property, assuming F is smooth ...
I am quite confused probably on the notation
but basically
surreal ember
You can handwrite it and take a photo if that's easier for you.
loud goblet
F'_x(x,y) = F'_x(2a-x,2b-y)
surreal ember
Hm, not quite correct.
loud goblet
oh wait
surreal ember
Notice how the first argument on the RHS is 2a - x, right? You forgot to use the chain rule.
loud goblet
so F'_x(x,y) = F'_x(2a-x,2b-y) *(-1)
surreal ember
Yeah, nice!
Now, substitute (x, y) = (a, b) on both sides and you'll see something interesting.
surreal ember
Yeah, nice!
Again, this works only if F is smooth, of course.
But if the derivatives of F exist, then they must be 0 at the center.
loud goblet
okay makes sense, yes!! so we basially just showed that point (a,b) is the centre of K if and only if we meet two conditions: F(2a-x,2b-y) = F(x,y) ; F'x(a,b) = 0 and F'y(a,b) = 0
and basically what we did to F(x,y) is manipulate it with translating and symmetry
surreal ember
Well, only the first condition is necessary and sufficient.
loud goblet
oh
surreal ember
The second is more like a property.
surreal ember
Well, around (a, b), not necessarily the origin.
loud goblet
so if its not equal, it probably means it is not symmetrical around the origin?
oh
wait let me rephrase it sorry
surreal ember
For example, take an ellipse: (x - 1)^2 + 2(y + 1)^2 = 1. Clearly, it's symmetric around (1, -1), so you can check whether the conditions apply.
loud goblet
For example, take an ellipse: (x - 1)^2 + 2(y + 1)^2 = 1. Clearly, it's symmetri...
so you're basically showing with the property if (a,b) is a centre ( of symmetry) of some curve K
because I just did everything with desmos and yeah you basically first translate it to the origin then make use of symmetry and then again de translate it
so you have the same ellipse in this case
surreal ember
Yeah.
And if there doesn't exist a point (a, b) such that F(2a - x, 2b - y) = F(x, y) for all (x, y) in the domain of F, then the curve is noncentral. Think parabola, for example.
loud goblet
okay makes sense, and just another question why did we use symmetry though to come up with 2a-x and 2b-y
surreal ember
Well, a center is an element of symmetry. The expressions are just what comes out when you invert around (a, b).
loud goblet
hmm I see
thank you very much btw!
I have a few other questions if you don't mind
We say that L is a tangent line to M in point p <=> the direction vector of L is perpendicular to the gradient of F(p)
surreal ember
Right, yeah.
surreal ember
Ah, for this you need the following useful property:
If you have a curve F(x, y) = 0, then ∇F at any point on this curve is perpendicular to this curve.
Using that, the property above is obvious - if the gradient is perpendicular to the curve at a point, then it's also perpendicular to the tangent line at that point.
surreal ember
Nice!
surreal ember
Ah, for this you need the following useful property:
If you have a curve F(x, y)...
By the way, this also works for surfaces F(x, y, z) = 0.
That can be quite useful in physics, for example, when working out field lines from equipotential lines/surfaces or vice versa.
loud goblet
By the way, this also works for surfaces F(x, y, z) = 0.
oo I see makes sense yeah
pine cypress
The property that the gradient is perpendicular to the level curve is also the core of the proof for what's known as a Lagrange multiplier
It's a concept used to find minimal and maximal values of a function
On the boundary of a domain
loud goblet
The only thing that still doesn't make any sense after the previous post was that that the tangent line T is tangent to a curve on a point <=> T intersects K multiple times in point p but I guess im too weak to understand that
loud goblet
It's a concept used to find minimal and maximal values of a function
ye I think we'ill see it later not sure
surreal ember
The only thing that still doesn't make any sense after the previous post was tha...
Well, it's not very good to say it exactly like that.
More like to what order the curves coincide at a point.
For two curves to be tangent to each other they must coincide at least up to first order; that is, if the curves are r1(t) and r2(t), then they are tangent at t = 0 if r1(t0) = r2(t0) and r1'(t0) = r2'(t0).
loud goblet
but aren't we just talking about one curve here
surreal ember
So, both their values coincide and their tangent vectors are parallel.
surreal ember
but aren't we just talking about one curve here
Well, the tangent line is the second curve in this case.
surreal ember
This case is more general; for example, two circles (x - 1)^2 + y^2 = 1 and (x + 1)^2 + y^2 = 1 are tangent to each other at the origin.
surreal ember
For two curves to be tangent to each other they must coincide at least up to fir...
Oh, though, one thing: the second identity with derivatives can be satisfied up to a constant, so r1'(t0) = const*r2'(t0). We want the tangent vectors to be collinear, not necessarily identical, after all. The first identity must hold, though, since otherwise the curves don't even meet.
loud goblet
yeah
but what did they exactly mean with T intersect curve K multiple times in point p
surreal ember
Well, saying it like that isn't very good.
loud goblet
yeah thats why I was so confused
surreal ember
By that, they mean specifically that the curves must coincide up to at least first order.
That "multiple" is the order.
loud goblet
hmm
surreal ember
Have you encountered Taylor series before?
For example, saying that sin(x) = x - x^3/6 + o(x^3) and cos(x) = 1 - x^2/2 + o(x^2) for x -> 0.
loud goblet
yes, but it has been quite a while
surreal ember
Well, it's kinda the same idea.
For example, sin(x) and x - x^3/6 coincide up to the third order, and cos(x) and 1 - x^2/2 coincide up to the second (and third, actually) order.
loud goblet
yeah
but essentially from what I remember is that a tangent line is basically a line that just touches the point and so it kind of intersects one time and after that its still very near to it but it isn't touching it
and now they kinda switched the definition to the above
surreal ember
Well, the more precise way to define it is as above: same value as the curve and parallel to the tangent vector of the curve.
loud goblet
but I was looking at some exercise and we had to show that the point p is an intersection point but you'ill get it two times and not just one
that's why I kind of wanted to understand that "weird" definition
surreal ember
but I was looking at some exercise and we had to show that the point p is an int...
Think of it this way: how many roots does f(x) = x^2 have?
loud goblet
And here it's sort of similar.
how?
surreal ember
how?
Well, instead of multiplicities of roots, here it's how many orders the curves coincide by, as described above.
loud goblet
Well, instead of multiplicities of roots, here it's how many orders the curves c...
Could you maybe show visually what that means?
surreal ember
Could you maybe show visually what that means?
Sure!
For example, here are the cases where the curves coincide roughly up to zeroth, first and second order. What that specifically means is that if you start at the intersection point and move by distance t from that point on each curve, then for small enough t the distance between resulting points scales as some constant multiple of t (first picture), t^2 (second picture) and t^3 (third picture).
So, the greater the order, the slower the curves diverge from each other when moving away from the intersection point.
The order can potentially be as high as you want, of course, assuming the curves are smooth enough.
loud goblet
Sure!
For example, here are the cases where the curves coincide roughly up to ze...
I see where you're going but, it won't be a tangent line anymore for 2 it's pretty obvious
surreal ember
I see where you're going but, it won't be a tangent line anymore for 2 it's pret...
Well, as I said above, tangent curve needs to be of order at least 1. A tangent line is of order exactly 1.
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loud goblet
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