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Linear algebraAnswered question
Modelfino0g Modelfino0g 2022-09-05

Given ax+by+c=0, what is the set of all operations on this equation that do not alter the plotted line?
Operations such as f ( x ) + a a, are obvious candidates for such a set. However, e.g., for the line y=−x, it seems to me to be non-trivial that x 3 + y 3 = 0 will plot the same line but x 2 + y 2 = 0 won't. Translation between coordinate systems also seems to be a non-trivial example. Is there any way to designate such a set? (Could this be generalized to other types of curves?)
The following are some more thoughts on the question:
It would be interesting in order to find alternate equation forms that might make more clear certain properties of a curve. For instance, x a + x b = 1 makes immediately obvious the abscissa and ordinate at origin. But we know that under some types of algebra, a x + b y + c = 0 might fail to be represented x a + x b = 1. So we're lead to think that these two equations plot a line by virtue of legitimate operations between them.
The equation of a plane also seems to be nicely related to the general form of a line, if r 0 = ( x 0 , y 0 ) and r=(x,y) are two vectors pointing to the plane and the normal is n = ( n x , n y ). If between vectors is the dot product, ( x x 0 , y y 0 ) n = ( x x 0 ) n x + ( y y 0 ) n y = n x x + n y y ( x 0 n x + y 0 n y ) = a x + b y + c = 0
The idea is to be able to see how the form of an equation can be altered, not the content of the variables. It seems odd to me that very complicated equations could have the same plotted curve as simple forms, but that this property wouldn't appear by virtue of the equation themselves, or the set of valid operations on this equation. This might seem weird, but say it is never immediately obvious that ax+by+c=0 plots a line, or x 2 + y 2 = r 2 plots a circle, unless we actually do the plotting, and ax+by+c=0 seems way less fundamental than y=mx+b.
Note that in the case of a circle, we have the pythagorean theorem that seems to be its clearest representation with the methods of analytic geometry, and the moment an equation can be said to share some sort of operation set with the pythagorean theorem, we know we're speaking of a circle. It seems that if we could somehow draw the operation set of a circle, we would get something like the pythagorean theorem, and that this operation set gets somehow deformed in order to give a representation onto the cartesian plane. For a translated circle with center (h,k), x 2 2 x h + h 2 + y 2 2 y k + k 2 = r 2 means absolutely nothing to us, but the form ( x h ) 2 + ( y k ) 2 = r 2 is clear as day.

Linear algebraAnswered question
manudyent7 manudyent7 2022-09-05

Trouble with the definition of the cross product
I am trying to understand the definition of the cross product given by Wikipedia
The article says that we can define the cross product c of two vectors u,v given a suitable "dot product" η m i as follows
c m := i = 1 3 j = 1 3 k = 1 3 η m i ϵ i j k u j v k
To demonstrate my current understanding of this definition, I will introduce some notation and terminology. Then I will show where my confusion arises with an example. I do apologize in advance for the length of this post.
Let M be a smooth Riemannian manifold on R 3 with the metric tensor g. Pick a coordinate chart (U, ϕ) with ϕ a diffeomorphism. We define a collection β = { b i : U T M | i { 1 , 2 , 3 } } of vector fields, called coordinate vectors, as follows
b i ( x ) := ( x , ( δ x ϕ 1 q i ϕ ) ( x ) )
where δ x : R 3 T x M denotes the canonical bijection. The coordinate vectors induce a natural basis γ x at each point x U for the tangent space T x M. Let [ g x ] S S denote the matrix representation of the metric tensor at the point x in the standard basis for T x M and let [ g x ] γ x denote the matrix representation in the basis γ x .
My understanding of the above definition of the cross product now follows. Let u , v T x M be tangent vectors and let
[ u ] γ x = [ u 1 u 2 u 3 ]             [ v ] γ x = [ v 1 v 2 v 3 ]
denote the coordinates of u,v in the basis γ x . Then we define the mth coordinate of the cross product u × v T x M in the basis γ x as
( [ u × v ] γ x ) m := i = 1 3 j = 1 3 k = 1 3 ( [ g x ] γ x ) m i ϵ i j k u j v k
Now I will demonstrate my apparent misunderstanding with an example. Let the manifold M be the usual Riemannian manifold on R 3 and let ϕ be given by
ϕ ( x 1 , x 2 , x 3 ) = ( x 1 , x 2 , x 3 x 1 2 x 2 2 )
ϕ 1 ( q 1 , q 2 , q 3 ) = ( q 1 , q 2 , q 3 + q 1 2 + q 2 2 )
The Jacobian matrix J of ϕ 1 is
J = [ 1 0 0   0 1 0   2 q 1 2 q 2 1 ]             J 1 = [ 1 0 0   0 1 0   2 q 1 2 q 2 1 ]
And the matrix representation of the metric tensor in the basis γ x is
[ g x ] γ x = J T [ g x ] S J = [ 1 + 4 q 1 2 4 q 1 q 2 2 q 1   4 q 1 q 2 1 + 4 q 2 2 2 q 2   2 q 1 2 q 2 1 ]
Now choose x = ( 1 , 1 , 1 ). The coordinates of x are evidently ϕ ( x ) = ( 1 , 1 , 1 ) and the three matrices above become
J = [ 1 0 0   0 1 0   2 2 1 ]             J 1 = [ 1 0 0   0 1 0   2 2 1 ]             [ g x ] γ x = [ 5 4 2   4 5 2   2 2 1 ]
Now we compute the cross product in the basis γ x . Using my understanding of the definition as outlined above, I get
[ u × v ] γ x = [ 36   35   16 ]
If we instead compute the cross product in the standard basis, then using my understanding of the definition, I get
[ u × v ] S = [ 0   1   2 ]
Naturally, these results ought to agree if we perform a change of basis on [ u × v ] γ x . Doing just that, I get
[ u × v ] S = J [ u × v ] γ x = [ 1 0 0   0 1 0   2 2 1 ] [ 36   35   16 ] = [ 36   35   158 ]
Clearly, these do not agree. I can think of several reasons for this. Perhaps the definition given on Wikipedia is erroneous or only works for orthogonal coordinates. Perhaps I am misinterpreting the definition given on Wikipedia. Or maybe I have made an error somewhere in my calculation. My question is then as follows. How should I interpret the definition given on Wikipedia, and how should one express that definition using the notation provided here?

Linear algebraAnswered question
Zackary Duffy Zackary Duffy 2022-09-04

Deriving the distance of closest approach between ellipsoid and line (prev. "equation of a 3-dimensional line in spherical coordinates")
Currently trying to solve a problem of calculating the smallest distance between a given ellipsoid centered on the coordinate system starting point and a given line (located... somewhere).
After chasing a few promising but non-functional methods I have settled on trying to use spherical coordintates - to determine the formula for said distance by determining the formula of the distance of the ellipsoid from the center and doing the same for the line, consequently subtracting the two, and using gradient descent on the resulting function to approach a minimum (hopefully the global one).
However, while that makes the ellipsoid calculations easy, I have found no consise way of determining a line in spherical coordinates in equation form. I have found the old questions on similar topics proposing use of Euler angles and the like, but that does not seem to be the solution (possibly because I haven't managed to appreciate it). So, asking here - is there any way to derive an equation for a line in 3-dimensional space in spherical coordinates?
Alternate methods for the task at hand are appreciated too - for the record, my previous lead was using a cyllindrical coordinate system with the line as its X axis, but the resulting formula for the ellipsoid turned out to be bogus.
Edit: may have figured out a solution for the bigger problem that does not rely on the spherical equation - see my own answer to the question. Title changed accordingly.
Edit 2: Scratch that. Fell into the same trap again; that solution is not going to work.

Linear algebraAnswered question
Darius Nash Darius Nash 2022-09-04

Polar Coordinate Transformation - Motivation
I am trying to work out the reason why the integral
e ( x 2 + y 2 ) d x d y
is, in polar coordinates,
e r 2 r d r d θ
As I understand it, a polar coordinate transformation involves the following substitution:
( x , y ) ( r cos θ , r sin θ )
This would imply that
( x 2 + y 2 ) = ( ( r cos θ ) 2 + ( r sin θ ) 2 ) = r 2 ( ( cos θ ) 2 + ( sin θ ) 2 ) = r 2
which gets us this far
e r 2 d x d y
To motivate why
d x d y = r d r d θ
I thought of the following argument:
d x = d ( x ( θ , r ) ) = ( x ( θ , r ) ) θ d θ + ( x ( θ , r ) ) r d r + 2 ( x ( θ , r ) ) ( r ) 2 ( d r ) 2 + . . . = r ( sin θ ) d θ + d r cos θ + . . .
d y = r ( cos θ ) d θ + d r sin θ + . . .
d x d y = [ r ( sin θ ) d θ + d r cos θ + . . . ] [ r ( cos θ ) d θ + d r sin θ + . . . ] = r d r d θ ( ( cos θ ) 2 ( sin θ ) 2 ) + . . . = r d r d θ ( ( cos θ ) 2 ( sin θ ) 2 ) ,  ignoring  o ( ( d r ) 2 )  and  o ( ( d θ ) 2 )  terms.
However, I am off by a minus sign, which would enable me to argue
d x d y = r d r d θ ( ( cos θ ) 2 + ( sin θ ) 2 ) = r d r d θ
If it were correct, I would find this line of argument would be much more analytically convincing than the typical argument involving
d x d y = d A = r d r d θ
which I find to be less mechanistically obvious than the above substitution-based argument that I considered but that I am not being able to fully justify.
Could you please tell me whether my substitution-based argument can work, potentially by correcting some mistake or another that I might have made? If not, do you have any similarly analytical or mechanistic justification as to why d x d y = r d r d θ ?


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